Weil's Representation and the Spectrum of the Metaplectic Group

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

530 Stephen S. Gelbart

Weirs Representation and the Spectrum of the Metaplectic Group

Springer-Verlau Berlin.Heidelberg-NewYork 1976

Author Stephen Samuel Gelbart Department of Mathematics Cornell University Ithaca, N.Y. 1 4 8 5 3 / U S A

Library of Congress Cataloging in Publication Data

Gelbart, Stephen S 1946Well's representation and the spectrum of the metaplectic group. (Lecture notes in mathematics ; 530) Bibliography: p. Includes index. 1. Lie groups. 2. Linear algebraic groups. 3. Representations of groups. 4. Automorphic forms. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 530. QA3.I28 no. 530 [QA387] 510'.8s [512'.55] 76-45609

AMS Subject Classifications (1970): 10 D 15, 22 E 50, 22 E55

ISBN 3-540-07799-5 Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-38?-0?799-5 Springer-Verlag New York 9 Heidelberg 9 Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.

For Mary

CONTENTS Introduction w

Background

w

Metaplectic

w

w

w

w

and

summary

groups

and

2.1.

Local

2.2.

Global

theory

2.3.

Well's

metaplectlc

2.4.

A philosophy

2.5.

Extending

2.6.

Theta-functions

AutomorDhic

Connections

3.3~

The

3.4.

Odds

11

theory:

13

representation

to

28

. . . . . . . .

39

GL 2 . . . . . . . .

41

. . . . . . . . . . . . . . . . . . . .

with

the

classical

of a u t o m o r p h i c

KrouD

. . . . . . . . .

theory

. . . . . . . . .

46 51 51

forms . . . . . . . . . . .

of the m e t a p l e c t i c

ends

22

. . . . . . . . . . .

representation

on th~ m C t a o l e c t i c

spectrum and

representation

for W e l l ' s

forms

Factorization

group

58

. . . . . . . . .

62

. . . . . . . . . . . . . . . . . . . . . .

69

arehimedean

places

. . . . . . . . . . . . . .

72

representation

theory

. . . . . . . . . . . . . .

72

4.1.

Basic

4.2.

The

4.3.

Application

4.4.

The b a s i c

local

theory:

map

. . . . . . . . . . . . . . . . . . . . .

of W e i l ' s

Well

representation

representation

the p - a d i c

96

representation

Class

i representations

5.3.

Hecke

operators

5.4.

The

5.5.

The b a s i c theory

Well

theory

. . . . . . . . . . . . . .

96

. . . . . . . . . . . . . . . .

102

. . . . . . . . . . . . . . . . . . . .

I05

. . . . . . . . . . . . . . . . . . . . .

111

representation

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

6.1.

The

discrete

6.2.

Construction

6.3.

Open p r o b l e m s

86

places . . . . . . . . . . . . . . .

Basic

map

81 .

93

5.2.

local

in 3 - v a r i a b l e s

. . . . . . . . . . . . .

5.1.

Global

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

Well's

3.2.

Local

representations

1

theory . . . . . . . . . . . . . . . . . . . . . .

3.1.

Local

of r e s u l t s . . . . . . . . . . . . . .

non-cuspid~l of cusp

forms

spectrum . . . . . . . . . . . of h a l f - i n t e g r a l

weight.

. . . . . . . . . . . . . . . . . . . . .

118 121 122 125 132

References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

Subject

138

Index

. . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction These notes are expanded from my earlier paper graphed at Cornell in July 1974.

[8] mimeo-

In addition to including correc-

tions and revisions for [8], the present notes contain new results and insights obtained during the past year and a half.

Some of

these results have already been described briefly in [9] and [i0]. Others are due to Roger Howe and Pierre Cartier. parts of Subsections pondence with Howe,

In particular,

2.4, 5.5, and 6.2 are taken from corresand parts of Subsections 2.5 and 3.1 were sug-

gested by Cartier after his critical reading of the original manuscript.

My indebtedness to both goes beyond acknowledgement

of

their suggestions within the text. The goal of these Notes is a general theory of automorphic form for the metapleetie group.

I am indebted to Robert Langlands

for inspiring this project and giving freely of his ideas. also grateful to Paul Sally for his collaboration Subsections 4.4 and 5.5), Kenneth Brown,

and my colleagues

Robert Strichartz,

on [i0]

at Cornell,

I am (cf.

especially

and William Waterhouse,

for

many helpful conversations. The theory of automorphic

forms on the metaplectic group

described in these Notes is still in its infancy. of the results,

if not incomplete,

Moreover,

are in preliminary form.

many I am

grateful to all the people named above for helping me realize this.

The expeditious typing of these Notes was done by Joanne Lewis, Arletta Havlik and Esther Monroe. S. Gelbart November 1975

B a c k ~ r g u n d a n d ,Sum/nar~ o f R e s u l t s .

w

The m e t a p l e c t i c group was f i r s t

i n t r o d u c e d by Well i n [47].

His purpose was to reformulate Siegel's analytic theory of quadratic forms in group theoretic terms~

The motivation for our investigation

comes from more recent works of Shimura and Kubota.

Our purpose

is to describe the spectrum of the metaplectic group modulo its subgroup of rational points and to relate this spectrum to the theory of automorphic forms for

GL(2).

Our work relies heavily upon Weil's. suggested,

However,

as already

it is more closely related to important recent discoveries

of Shimura and Kubota which we shall now briefly describe. Fix

k

to be an odd positive integer,

divisible by 4, and N.

X

N

a positive integer

a character of the integers defined modulo

Put

to(N)

=

{[~ bd] ~ sL2(~):

c

9

O(N)},

and e(z) =

~

exp(2vi n2z).

Then

le(Yz)/e(z)i for all

Y c to(N).

= [cz+dj I/2

(Hecke [16], pp. 919-940.)

In [38] Shimura deals with cusp forms

f(z)

satisfying the

identity

f(~z) = ~(d) E~(~z)/6(z)]kf(z) for all

F ~ ~o(N).

cus~ forms of weight [~(u

Such functions comprise the space of classical k/2, character

-I/2.

X, and "6-multiplier system"

We denote this space by

Sk/2(N,X ).

Func-

tions in it arise naturally in number theory from the study of partitions and quadratic forms in an odd number of variables.

To study functions in N

operators

Tk,x(m )

weight).

These

However,

~,~(m)

square and

Sk/2(N,X )

one introduces certain linear

(following Hecke in the case of forms of integral

T(m)

operate in

Sk/2(N,X)

for all integers

operates as the zero operator if

(m,N) = 1.

m

m.

is not a

This curious fact seems to have discouraged

Hecke from beginning a systematic study of such forms along the lines of his already successful theory for forms of integral weight. theless,

Shlmura established

Suppose

the following provocative

f(z) = Z a(n)exp(2~i n z)

eigenfunction

for every

T N x(p2), k,

in

result.

Sk/2(N,X )

is an

say

T(p2)f : ~,(p)fo Suppose also that

k > 3o

Then

n:l

.

= ~[l_~(p)p-s P where to

L*(X,.)

- - ~ - -s) + ~(p)2pk-2-2s]-l,

is essentially the Dirichlet L-series associated

y; furthermore,

and more significantly,

the inverse Mellln

transform of A ( n ) n -s : H [ l - k ( p ) p -s + X(p)2pk-2-2S] -I,

i

p

namely F(Z) =

~ A(n)exp(2vi n z), n=l

satisfies

F(~z)

for all

y e ~0(No).

usually equals

N/2. )

= X(d)2(cz+d)k-iF(z),

(Here

NO

depends only on

N

and

X

and

None-

The signlflcance

of Shimura's result is that it establishes

a correspondence between forms in (actually even) weight

k-1.

Sk/2(N,X)

Note that

and forms of integral

T(p)F = k(p)F.

Thus this

correspondence preserves eigenvalues for the Hecke ring. The success of Shlmura's theory leads one to ask several important questions.

For example,

can Shimura' correspondence be

defined without recourse to L-functions? group theoretic interpretation?

In particular, what is its

Is the correspondence one-to-one?

]~qthe first part of this paper I shall interpret forms of halfintegral weight as irreducible representations of the metaplectlc group "defined over

Q."

Thus, the full weight of the representation

theory of this group can be used to discuss these questions. Kubota's results also concern forms of "half-lntegral weight". To describe them, fix an imaginary quadratic field F =

denote by

0

the ring of integers of

power residue symbol in subgroup mod r(N)

Q(J'Z'd),

N

of

F.

SL(2,0)

Let

F,

F(N)

and by

(~)

the quadratic

denote the congruence

and define the function

X(y)

on

by

1

otherwise .

The starting point for Kubota's investigation is his discovery that

~

is a character of

congruent to

0

law for

modulo the fact that

(See [19]

(~)

mod 4).

F(N)

(provided

N,

as before,

is

This result is equivalent to the reciprocity ~d

is totally imaginary.

for the original proof and Subsection 2.2 of this paper

for the case of arbitrary number fields. Now let

H

denote the three-dimensional quaternionic upper

half-space whose points are of the form (z e C, v > 0).

The operation of

Z u = (z,v) = iv

~ = [c a bd ] ~ SL(2,~)

-V

~]'

on

H

is

given by

% (identifying to

t e ~

SL(2,~)/SU(2)~

-- (au+b) (cu+d)-l

with the matrix The quotient

[0

])

so

H

is isomorphic

space

r(~)/H is of finite

volume.

In [21] Kubota to

P(N)

considers

and character

X;

E(u,s)

modular

function~

his special

:

Z

on

interest

•

H

with respect

is in

s+l

r XZ(Ni where F|

s

is a complex

is the upper The series

variable,

triangular defining

only for

continuation

to the whole

for all

form which

s-plane

%(u)

of

over

Rather

~(~T~.

One of Kubota's

u = (z,v),

E(u,s)

series.

and a pole

of the first order at

at

s = 1/2

~

a function

the metaplectic

satisfies

automorphic

defines

a theta-function. of

on a certain

GL(2) two-fold

group of w

results

is a computation

of ~(u)

generalizes

the fact that the eigenvalues

of the classically

holomorphic

Eisenstein

) O]

arithmetic

functions

As already

in

to Hecke's

Jim(z)

ring.

of the

elgenvalues

series

one purpose

This result defined

are given by

such as the sum of the divisors

suggested,

Although

itself has an analytic

to the adele group

with respect

and

F(N).

In fact

it defines

principal

if

a s~are-inte~rable

be lifted

of this group,

of

E(u,s)

is not a ~usp form~ can not

= v

is an Eisenstein

~ l,

and defines

This function

covering

Re(s)

The residue

~ e ~(N)

subgroup

E(u,s)

it converges

s = 1/2.

v(u)

of an integer.

of our investigation

is

to reformulate

and extend the results

of Shimura and Kubota in the

context

of a general theory of automorphic

forms for the metaplectlc

group.

In the context of this general theory,

and Kubota appear to be closely related. shed llgat on both of them~ a tensor product local groups computation

Gv

Kubota's

"extraordinary"

@(u)

becomes

representations

defined at each place of

of eigenvalues

of Shimura

Thus it is hoped we have

In particular,

of certain

the results

F.

of the

The corresponding

then becomes part of the spherical

function

theory of these groups. In addition to working over arbitrary ntumber fields, are new.

our methods

In that our point of view is representation-theoretic

follow Jacquet-Langlands

([18]) rather closely.

we

In fact, a second

gaal of cur program is to develop a theory for the metapleetlc

group

analogous

GL(2).

to Jacquet-Langlands'

The possibility

treatment

of Hecke theory for

of such a theory was already

suggested by Shimura in

[38]. To be more precise, and

~

let

F

its rang of adeleso

denote an arbitrary

Let

G

denote

algebraic

group over

F.

The meta~lectic

extension

of

by

Zn,

GLn(~ )

aL 2

group

global field

regarded as an G~

is a central

the group of square roots of unity.

Thus

1

i s e x a c t and

Z2

Gin(F)

~atur.a.l unitary

Gs2(

is a trivial

of this extension points

s2

)

1

GL2(#)-module.

The c r u c i a l

is that it splits over the subgroup

and the fundamental representation

of

property

of rational

problem is to decompose the G~

in

Ln(GF~).

Suppose we denote this representation

by

~o

Then

T(gO)~(g) = ~ ( g go ) for all

g,

gO ~ ~ '

~d

m ~ L2(GF~)

~

Note t~at

L2(GF\~),

6

L2(G~"~,~A/Z2)__ (which

as a T - m o d u l e ,

is the direct sum of

L2(GF'~DL2(~)))

and the space of functions

~(~g) : ~ ( g ) , ("genuine" functions on

~)o

--~s

)

is Just

satisfying

~ e Z2)

Thus

T=T| where constituents of

T

correspond to aut0morphic forms on

The constituents of Langlands'

T

GL(2J.

comprise the subject matter of Jacquet-

treatment of Hecke's theory of forms of integral weight.

The irreducible constituents of

T,

on the other hand, are what we

call "genuine automorphic representations of

the metaplectic group",

or "generalized automorphic forms of half-integral weight over

F".

This terminology is apt since the forms considered by Shimura correspond to special forms on

~

defined over

is the starting point for our theory;

Q.

(This observation

see Subsection 3.1 for details.)

The Eisenstein series considered by Kubota lead to forms which are defined over a totallz ima~inar[ field

F,

and which occur outside

the space of cusp forms~ The main emphasis of our theory is on relations between automorphic forms on the metaplectic group and automorphic forms on GL(2), i.e. between constituents of

~

Eventually we shall define a map

between arbitrar[

necessarily automorphic) and representations of

S

and constituents of

representations of GA.

~

(i.e. not

non-trivlal on

Z2

This map will be consistent with

ShimuraTs when restricted to automorphic forms on it will be one-to-one,

T.

G~.

Moreover,

and its definition will be entirely represen-

tation theoretic. A correspondence and

~

D3

between certain representations of

is constructed completely independently of

theory of the metaplectic group. S. Niwa

[30] and T. Shintani

S

GA

using Well's

Motivated by recent results of

[hl] we collect evidence for the

assertion that

D3(s(w)) whenever

S(~)

=

is in the domain of

D3o

The validity of this

h y p o t h e s ~ is one of the focal points of our theory.

We also describe

other features of our theory and explain its connection with the results of Shimura and Kubotao

All our results lend support to the

assertion that forms on

and the metaplectic

GL(2)

group are inex-

tricably connected. A more precise summary of the contents of this paper now follows. In Section 2 we analyze the metaplectic over local and global fields. follow Kubota,

then Weil~

coverings of

GL(2)

In constructing these groups, we first

Whereas Kubota's construction involves an

explicit factor set and is well-suited

for basic computations,

construction is entirely representation-theoretic larly crucial to our approach.

In Subsections

Weil's

and hence particu-

2. h and 2.5 we explain

how Well's construction leads to a general philosophy which not only yields the map

D

alluded to above but also a correspondence

between automorphic forms on

~

and ~utomorphic forms on

GL 1.

These matters are dealt with in detail in Sections 4,5, and 6. speaking,

to each Well representation

quadratic form Dq

q

and to each

between irreducible

q

r

of

~

r 9 q

constituents of

r

q

In particular,

q3(xl,x2, x3) = x I yields the map

D3

alluded to above, while

ql(x) relates to Kubota's results and

Roughly

there corresponds

a

there is attached a correspondence and constituents

natural representation of the orthogonal group of of

D1

= x2

DIo

q

of the

in the space

In Section 3 we introduce paper.

the subject matter proper

Some general features of the decomposition of

are sketched and the connections between constituents

of this

L2(GF~) of this

decomposition and the functions considered by Shimura and Kubota are described in detail.

Some miscellaneous

used later on are also collected here.

results which will be

Our basic observation in

Section 3 is that one can make sense out of the relation ~=|

for certain irreducible that

GA

"genuine" representations

~.

(Note

This result is important since it reduces

global questions to local ones. S

of

will not be a restricted direct product of the local

covering groups ~v. )

map

V

alluded to above,

In particular,

in describing the

one is lead to the study of certain local

maps Sv: ~v ~ ~v for each place

v

of

F.

cuspidal spectrum of series on

In this

L2(GF~)

Section we also isolate the non-

using the theory of Eisenstein

G~o

In Section 4 we treat the local map for the archimedean places in complete detail. of

Fv = ~

as

Gv

the map

or

~,

To certain pairs of quasi-characters irreducible representations

are described. Sv

If

~--v is attached

is defined by setting

representation

of

Gv

attached to

Sv(~v ) (~12,~)o

to

of

Gv

(~i,~2)

lowest weight Gv

as well then

equal to the irreducible This is consistent with

Shimura's map since a discrete series representation of k ~

(~I,~2)

~v

with

is mapped to a discrete series representation of

with lowest weight

k-lo

(It is consistent with Kubota since it

attaches the trivial representation of series representation of index

G@

s = i/2o)

to the complementary

In Subsection attached

4.4 we describe

to the quadratic

form

the problem of decomposing forms in an odd number it deserves.

This involves attached

the decomposition form

GR

S v. G~

of a map

In particular, corresponds

q3(xl~x2,X3)o

~k-i ~ ~i/2

inverse

to

S v.

r3

The orthogonal

group r3

of a certain regular representation to (most)

and t~as one obtains

to the discrete

of

representations an inverse

series representation

series representation

This result seems to be new~

result in

and the idea is to decompose

~

the discrete

to quadratic

of the Well representation

GR

of

In general,

attached

D~

The result is that one attaches

a representation

~.

The most significant

the construction

according to the decomposition

of

over

rI

has not yet received the attention

[52].)

of this form is essentially

itself~

decomposes

of variables

to the quadmatic

G~

q!

Well representations

(See, however,

Section 4 concerns

how the Weil representation

to

~k-i

~k/2

of

of

~.

A classical version of the correspondence

(using theta functions

was first obtained by T~ Shint~li

in place

of Well's

representation)

([41])o

In Section 5 we begin the local tDeory for non-archimedean places by describing tions

Sv

for a wide class of irreducible

(the non-supercuspidal

of quasi-characters

of

Fv

representations).

We also analyze

Hecke algebra~

showing that our map is consistent finite places preserved

as well,

i.eo,

(cf. Theorem 5.13).

the basic non-archimedean gether with the results correspondence

DI

we define

and inducing up to

the class 1 representations

for the generalized

After attaching pairs

to such representations

again by squaring these characters

representa-

and compute

These results

Sv

G v. eigenvalues

are useful in

with Shimura and Kubota at the

eigenvalues

for the Hecke ring are

In Subsection

Well representation

of Subsection

5.5 we describe rI

how

decomposes.

To-

~.h this leads to the global

alluded to above.

In Section 6 we attempt

to tie together the local threads

of

10 Sections

A and 5.

Our goal is a global description

of the space of automorphic

(genuine)

the theory of Well represenbations

forms on

expounded

locally with the theory of Eisenstein The result is a complete discrete non-cuspidal the implication

series

characterization

spectrum of

of the constituents

~.

Roughly speaking,

in Section 2 is mixed sketched

in Section 3.

in Subsection

L2(G~).

6.1 of the

In classical

is that any square-integrable

modular

terms

form of half-

integral weight which is not a cusp form must be a "translate" the basic theta-function be generated by residues kind of Siegel-Well In Subsection Here ignorance

6(z).

series,

(cf. [48] and

this amounts

essentially

dominates

6.3 describes

the situation.

global constituents

bute to the space of cusp forms~

Similar results

some speculations

to return to in future papers~

to a

[i0])~

6.2 we treat the cuspidal spectrum of

shown that all "non-trivial"

Subsection

Since such forms are also shown to

of Eisenstein

formula

of

of

L2(GF~).

Nevertheless rI

it is

indeed contri-

are discussed

for

and problems which we hope

r 3.

{2.

Metaplectlc

Groups and .Representations.

In [21] Kubota constructs of

GL2(g)

a non-trivlal

over a totally imaginary

number field.

I shall describe the basic properties arbitrary number field and complete struction

at the same time.

two-fold

covering group

In this Section

of such a group over an

some details of Kubota's

I shall also recall Well's

con-

construction

in a form suitable for our purposes. I start by discussing

the local theory and first collect

elementary facts about topological Let

G

denote a group and

as a trivial G-space. on

G

(2.1)

group extensions.

T

a subgroup of the torus regarded

A tvp-cocyc.le

is a map from

Ox G

to

T

some

(or multiplie.r,

or factor

set)

satisfying

~(glg2,g3~(gl, g2 ) = c(gl, g2g3)~(g2, g3)

and

(2.2)

~(g,e)

for all

g'gl

in

G .

=~(e,g)

in additlonl

if

= 1

G

is locally compact,

a

will be called Borel if it is Borel measurable. Following Moore 2-cocycles

on

G

cocycles

(cocycles

from

to

G

T).

dimensional represents G

by

T

~

let

and let

Z2(G,T)

B2(G,T)

of the form

equivalence

denote the group of Borel

denote

its subgroup of "trivial"

s(gl) s(g2)s(glg2 )-I

Then the quotient

cohomology

group of

G

group

H2(G,T)

with

a map

(the two-

with coefficients

classes of topological

s

in

coverings

T) groups of

which are central as group extensions.

To see this, class

[28],

in

let

H2(G,T). GX T

~

be a representative Form the Borel space

multiplication

in

by

(2.3)

[gl'~;1][g2'~;2)

= [glg2'~(gl'g2)r

of the cohomology GX T

and define

12

One can check that product for

G• T

of H a a r m e a s u r e s

G X T.

topology

G

and

invarlant

admits

a unique

compatible

with

the g i v e n

Borel

structure.

that

the n a t u r a l

from

locally

sequence

of

~/T

compact

class

of

The

with

~

measure

locally

and

latter map,

G.

that the

~

to

Borel,

compact

G

are

hence

moreover,

Thus we have

an exact

groups

as a group e

to

and

and o b v i o u s l y

continuous.)

is c e n t r a l

the e q u i v a l e n c e

g ~ (g, I)

T

(They are homomorphisms,

a homeomorphism of

maps

i ~ T -~ ~ - ~ G-~

~:

is an

G• T =~

they are a u t o m a t i c a l l y

This

T

group

[25]

continuous.

sequence

on

Borel

Thus by

Note

induces

is a standard

.

i .

extension

Its natural

and depends

Borel

only on

cross-sectlon

is

2. I. Local T heor[. Let

F

denote a local field oT zero characteristic.

is archimedean,

F

is

is a finite algebraic If of

F,

F U

or

extension

of

P .

Let

if

F

let P

0

denote

denote

F

Qp

the ring of integers

its m a x i m a l prime

q = lwl-I

F

is non-archimedean,

of the p-adic field

its group of units,

ideal,

the residual

and

characteristic

F . The local m e t a p l e c t l c

GL2(F) of

C;

is non-archlmedean,

a generator of

~q

If

which

involves

group

is defined by a two-cocycle

the Hilbert or quadratic

on

norm residue

symbol

F . o

The Hilbert F xx Fx

to

Z2

F(J~).

from

square.

(',')

which takes

(',')

Fx• Fx

to

(x,y)

itself

if

I

iff

x

in

is identically

is trivial on

Some properties use throughout

is a symmetric b i l i n e a r map from

(x,y)

In particular,

Thus

trivial on

sy~ibol

(FX) 2 • (FX) 2

Fx 1

Is a norm

if

y

is a

for every

F

and

F = ~ .

of the Hilbert

symbol which we shall repeatedly

this paper are collected below.

Proposition

2.1.

(2.41

(i) For each

(a,bl

= (~,-ab)

F,

(',.)

is continuous,

= (a,(l-a)bi

,

and

(2.5

(a,b)

= (-ab,a+b)

(ii)

If

q

is odd,

(u,v)

(ill)

If

q

is even,

and

then

(u,v)

is ident.lcally

1

is identically v on

The proof of this P r o p o s i t i o n of O'meara

[31] and Chapter

Now suppose or

d

according

c

in

U

is non-zero

i

on

is such that

U x U; vml(4),

U . can be gleaned from Section 63

12 of A r t i n - T a t e

ab s = [cd] 6 SL2(F) if

;

and set or not.

[i]. x(s)

equal to

c

14

Theorem 2.2.

(2.6)

The map

~: SL2(F) • SL2(F) ~ Z 2,

defined by

~(Sl, S2) :(X(Sl),X(S2))(-X(Sl)X(S2),X(SlS2) ) ,

is s Borel two-cocycle cohomologically

trivial

This Theorem preliminary

on

SL2(F ).

Moreover,

if and only if

this cocycle

F = @ .

is the main result of [20].

remarks

it determines

is

According

to our

an exact sequence of topological

groups

1 ~ z2 ~ s~2(F) ~ SL2(F) ~ I where

SL2(F)

is realized

plication given by (2.3). not the product suppose

N

The topology for

topology of SL2(F)

is a neighborhood

Then a neighborhood where

as the group of pairs

UeN

and

Proposition extension of

2. 3 .

SL2(F)

Z2

SL2(F)

F/@,

by

unless

is provided

is identically

If

SL2(F),

Z2

with multl-

however, F = @ .

basis for the identity

basis for

~(U,U)

and

[s,~]

in

is Indeed,

SL2(F).

by the sets

(U, 1)

one.

each non-trivial

is isomorphic

topological

to the group

SL2(F )

just constructed. Proof.

If

F =~,

nected Lie group, SL2(F)

isomorphic

obtained by factoring

on the other hand, shown

hence

each such extension

is that

that

F

is automatically

to "the" two-sheeted

its universal

cover by

is non-archlmedean.

H2(SL2(F),Z2) = Z 2.

a con-

cover of

2~.

Suppose,

What has to be

For this we appeal to a result

of C. Moore's. Let in

F .

EF

denote the (finite cyclic)

Consider

the short exact sequence

l~Z 2 ~ ~ / z The corresponding

group of roots of unity -

2~

l

long exact sequence of cohomology groups

is

15

'''~I(s~(F),~/Z2)

~2(SL2(F),Z 2) ~H2(SS2(F),S ;)

H2(SL2(F),EF/Z2) +H3(SL2(F),Z2) ~''' Recall that

HI(sL2(F),T ) =Hom(SL2(F),T).

its commutator subgroup. H2(SL2(F),Z 2)

Therefore

Moreover

. SL2(F)

equals

Hl(sL2(F),EF/Z2) =[1]

imbeds as a subgroup of

Theorem 10.3 of [29] it follows that

H2(SL2(F),EF).

But from

H2(SL2(F),EF) = E F .

deaired conclusion follows from the fact that

and

Thus the

H2(SL2(F),Z 2)

is

non-trlvial and each of its elements obviously has order at most two.

[] Remark 2.4.

of

SL2(F)

for

As already remarked, a non-trlvial two-fold cover F

a n on-archimedean field seems first to have

been constructed by Well in [47].

His construction, which we shall

recall in Subsection 2.3, is really an existence proof.

His general

theory leads first to an extension

where

on

T

L2(F).

is the torus

and

Mp ( 2 )

i s a group o f u n i t a r y

Then it is shown that

element of order two in Remark 2. 5 .

Mp(2)

operators

determines a non-trlvlal

H2(SL2(F),T).

In [20] Kubota constructs n-fold covers of

SL2(F ).

His idea is to replace Hilbert's symbol in (2.6) by the n-th power norm residue symbol in of unity).

F

(assuming

F

contains the n-th roots

In [29] Moore treats similar questions for a wider range

of classical p-adic linear groups. Now we must extend

~

to

a = as2(F) In fact we shall describe a two-fold cover of non-central extension of

SL2(F)

by

F x,

G

which is a trivial

i.e. a seml-direct product

16

of these groups. ab If g = [cd]

belongs to

G,

P(g) = [

ca

(2.7)

For

gl'g2

in

(2.8)

G,

write

i 0 g = [0 det(g) ]p(g)

where

~cSL2(F)

define

cz*(gl, g2) =ct(p(gl )det(g2) , p ( g 2 ) )

v (det(g2),p(gl))

where I 0 -I

(2.9)

I 0

sY = [0 y]

S[o y]

and

=f

(2. lO) if

v ( y , s)

s = [c

]'

Note t h a t

coincides with

e/O

otherwise

the restriction

If

{sY,~v(y,s)].

y c F x, Then

of

and

s-~s y

and the seml-direct product of isomorphic to the covering group Proof.

i~

c~~

to

SL2(F) X SL2(F)

a 9

Proposltion 2.6. equal to

1

\ (y,d)

~=[s,~]

c~2(F) ,

put

is an automorphism of

SL2(F)

and

~

G

of

Fx

~Y

SL2(F),

it determines

is

determined by (2.10).

It suffices to prove that

~ ( s l, s 2) = ~ ( s l Y , s2Y) v ( y , s 1) v ( y , s 2) v(y, s 1 s 2) and this is verified Remark 2.7.

in Kubota

[21] by direct computation.

We shall refer to

~

as "the" metaplectic

even though there are several (cohomologically extend [g,~]

~ with

to

G 9 g ~ G,

distinct)

group

ways to

We shall also realize it as the set of pairs ~ ~ Z 2,

and multiplication

described by

[gl,~l][g2,~2] = {gl, g2,~*(gl, g2)~l~2] Now let B,A,N, and K denote the usual subgroups of 9

Thus

G 9

17

A=

, ai~F

,

a2

a2

and

K

is the standard maximal

F =~,

0(2) If

inverse

if

H

F =R,

and

GL(2,0)

is an[ subgroup of

image in

~ .

isomorphic

to

H .

G,

Moreover,

is the direct product of

Z2

We shall denote

it is important

over the subgroups below

is useful

that if

x

by

~

splits over

H'

by

Proposition is a positive

H,

H'

H

even though

H'

H .

In particular,

field

~

splits

the Proposition

in constructing the global metaplectic

x=w~

then

of

to know whether or not

listed above.

if

will denote its complete

belongs to a non-archimedean

by the equation

G (U(2)

otherwise). H

if

subgroup of

and some subgroup

need not be uniquely determined In general,

compact

group.

(Recall

its order is defined

u eU.)

2.8.

Suppose

F/C

integer divisible

by

or h .

~

and

Then

N

~

(as usual)

splits over the

compact group =

More precisely,

ab [[c d ] 6K:

s([ c d]) -

Then for all

(2.12)

c ~0(mod

~)]

set

f(c,d

det(g))

if

cd~0

L

1

otherwise

and

ord(c)

is odd

gl, g 2 e K N ,

~*(gl, g 2) = S(gl)s(g2)s(glg2) -1

Proof. gl, g 2

ab g = [c d ] e G,

for

a b

(2.11)

a ~l,

in

Theorem e of [21] asserts ab [[c d ] e K:

ab [c d ] ~ 12(m~

that (2.12)

N)].

is valid for all

But careful

inspection

18

of Kubota's proof reveals that the conditions d i l(mod N)

are superfluous.

b--0(mod N)

and

Indeed Kubota's proof is computational

and the crucial observation which makes it ~ork is the Lemma below. We include its proof since Kubota does not. Lemma 2..9.

a b d ] c KN 9 k = [c

Suppose

s(k) =f(c,d(det k)) (2.13)

\

Proof.

1

and

d =0

-bc~U

c~0

~

[ac ~d]b

KN .

In particular,

U .

implies that

implies

c~U

otherwise.

belongs to

Suppose first that

c~U

c%0

Throughout this proof assume

dot(k) --ad-bc

Indeed

if

Then

and

c ~U

dot(k) =-bc,

Thus if

.

Then clearly

cd~0

.

a contradiction since

order (c) is odd,

s(k) = (c~d(det(k)) by definition.

On the other hand,

if

ord(c) = 2 n

(where

n~0

since

c~U)

it remains to prove that (c,d(det k)) = 1 . But

d(det k) = a d 2 - d b c ,

so

kcK N

implies that

d(det k) -~l(mod 4).

Thus (c,d(det(k))) = (w2nu, u ') = (u,u') = i by (iii) of Proposition 2.1. To complete the proof it suffices to verify that c~U

if

cd~0

and

since if

c cU,

then

ord(c)

is odd.

ord(c) =0,

c ~0

and

This is obvious, however,

a contradiction,

since

ord(c)

must be odd. Note that when the residual characteristic of

~ - - K : GL(2,0F).

F~ F

is odd,

19

Definitien 9.10. function on let

s(g)

G

KNX KN and

C,

let

s(g)

If

F

denote the is non-archimedean,

~(gl, gg)

denote

~*(gl,g2 ) s(gl) s(g2)s(glg 2) ~

determines a covering group of

G

isomorphic

But according to Proposition 2.8, its restriction is identically one.

KN

or

which is identically one.

Obviously ~ .

F =~

be as in (9.11), and in general, let

the factor set

to

If

Thus

lifts as a subgroup of

~N ~

is isomorphic to

via the map

k ~ {k,l].

this reason we shall henceforth deal exclusively with Lemma 9. II.

K~• Z 2 , For

~ .

Suppose

F

~i

gi =

xi] i=1,2

~ B,

o

Then

~(gl, g2) = (~i,~2) Proof.

Since

~i

xi

0

hi

=

0

~i

~i~DL 0

det(gi)

xi ~[l

P(gi)

,

it follows that

= (~{I,~I)(-WIIw2

-i

,

But using (2.4) together with the symmetry and billnearlty of Hllbert's symbol, this last expression is easily seen to equal Thus the Lemma follows from the identity

(Wl,~2).

20

(2.1~)

sIE~0 ~Jl = 1

valid for all

[

Corollary central in

~

Proof. ~.

Then

/3(y,y')

~] r B.

[]

2.12.

Suppose

iff y

is a square in

Suppose

Y = [~ 0].

V' = {Y',C']

Then

is

A. is a r b i t r a r y

= [[~0' O,],C,

V V' = Y' Y iff {YY',~(Y,Y')CC] = #3(y',y)

V = {Y,~

= [Y'Y,~(Y',Y)~']

in

iff

iff

(2.z5)

(~,~,)

= (~,,x) 4

So suppose

first that

are squares

in

one for all

Fx

y

and

by the n o n - t r i v i a l i t y

that

obtains by default

if

say,

= -i

(2.15)

Corollary

2.12'.

So does

~,

A2

2.13.

~' = I, say.

Thus

and the proof is complete.

The subgroup

N

of

G

The center of

z(g) (b)

F J g. ~

Then:

is

z o = {[[o z]'g]:

z ~

(F x) 2]

Suppose

-'2 G = [{g,(:] Then the center of

-J2 G

is

~'G:

det(g) c (FX) 2]

g = [[[~ 0z ] , ( ~ ] :

V

will not []

lifts as a subgroup

with

Fix

F x, then

h' ~ F x

Obvious.

Corollary (a)

and

symbol,

A 2 = {y ~ A: y = 8 2 , 6 ~ A] 9 Proof.

~

(both sides equal

is not a square in

for some

fails for

I 0 {[0 ~,],I}

commute with

~.

Then both

(2.15)

of Hilbert's

(~,k')

of

A.

~',k').

On the other hand,

This means

is a square in

z ~ F x]

9

21

if

Proof.

Since

g=[g,~]

eZ(~)

g = [~ oZ]

with

z eF x,

it suffices to prove that

commutes with ever[ Clearly

~(g,, [~ oz]),

only if

g' =([ca b ],~') e G

[[~ 0],~]

commutes with

iff g'

geZ(G),

i.e. only {[0z o],r z

z

is a square in iff ~([0z O z ]' g, ) =

Fx

But a simple computation shows that

a,(K

>=


and 0 ~*(g',[~ Z]) = (z,c)(z, det(g') So since s(g,)-ls([~ o

z o

z o

z o g,),

z])-Is(g'[o z])=s([ 0 z ] -Is(g' )-iS( [o z ]

0

{[~ z],(~} ~ Z(~)

(2.16) for all

if and only if

(z, det(g')) = i ~' ~ G,

[ [~ oz],~] c z(~) det(g') ~ (FX) 2, established.

or, since Hilbert's symbol is non-trivial, iff

z

is a square.

(2.16) holds for all

On the other hand, if z,

and hence (b) too is

[]

2.2. Global Theory. In this paragraph, v

a place of

adeles of

F, F v

F.

F

will denote an arbitrary number field,

the completion of

Thus the

G

F

at

v, and

of Section 2.1 becomes

its maximal compact subgroup is

K v, A

is

~

the

G v = GL2(Fv),

A v, etc.

Our interest

henceforth is in the global group G~ = GL 2 ( ~ )

and its two fold cover If if

g

and

g = (gv)

~v(g,g' ) = i

on

%•

(2.17)

g'

and

%.

are arbitrary in g' = (g~).

G~, put

~v(g,g') = ~(gv, g~),

According to Proposition 2.8,

for almost every

v.

Thus it makes sense to define

by

~(g,g')

= ~ ~v(g,g') v

the product extending over all the primes of obviously a Borel factor set on of

G~

pairs

which we shall denote by

GA ~A

F.

Since

~

is

it determines an extension and realize as the set of

[g,~], g e G~, ~ e Z 2, with group multiplication given by

{ g l , ~ l ] { g 2 , C2] = [ g l g 2 , ~ ( g l , Important subgroups of N

KO

=

G~

H

g2)cl~2 ].

include

KN

v<~

V

and GF = GL 2 ( F ) .

Proposition 2..!~.. The subgroup of

~

via the map Proof.

~0

of

k 0 ~ [ko, l]-

Proposition 2.8.

Proposition 2...15. For each

s~(u

= ~ v

u e GF, let

Sv(V)

G~

lifts to a subgroup

23

the p r o d u c t

extending over all

(finite)

primes v

of

F.

Then

the map

V ~ {V, s~(v) ] provides

an i s o m o r p h i s m b e t w e e n

Proof. 0

GF

Note first that if

c

is

for almost

v

and the product

every

Thus

To prove

(2.18)

s~(y)sA(y')/~(y,V') y, y'

So fix of

y

and

y y'

G F.

and

y'

in

2.1,

for all

is

Thus

~v(y,y')

= i

symbol is trivial of

v, all the entries

Fv

for almost all

on units if is odd.)

v On

= (rl, r2)v(rs,r~) v

r I, r 2, r 3, r 4 e P. symbol

sA(y )

v,

~v(y,u

for Hilbert's

every

suffice to prove that

For almost all

and the residual c h a r a c t e r i s t i c

the other hand,

with

it will

i.e.

of

= s~(YV')

G F.

Hilbert's

%"

for almost

is finite,

the P r o p o s i t i o n

will be units.

(By P r o p o s i t i o n is finite

in

Sv(Y ) : i

in (2.18)

well-defined.

for all

of

Y = [c a d b ] e GF, the v-order

v.

appearing

and a subgroup

Consequently,

(quadratic

v

~v(y,y,)

by the p r o d u c t

reciprocity

for

formula

F),

= 1 .

That is,

~(~,y' ) : s~(x)s~(~' ) s~(yx' ) as was to be shown.

[]

Because of this P r o p o s i t i o n homogeneous

where

Z 0oo

positive This

we can make

sense now out of the

space

denotes

the subgroup

real matrices

[~ 0].

of the center of (Cf.

Corollaries

space will be the focus of our a t t e n t i o n

G oo

,consisting of

2.12'

and 2.13. )

in Section 3.

v.

24

The P r o p o s i t i o n on

~

with classical forms

stating

it we collect

residue

symbol.

Suppose Let

below makes

S

denote

defined

some facts

a,b ~ F x

it possible

with

in the upper half-plane.

concerning

b

to relate functions

the quadratic

relatively prime to

the set of a r c h i m e d e a n

places

of

a

F.

Before

power

and

2 9

Suppose that

Ordv( b ) (b) = I1 v v where

(b) denotes

the F-ideal generated

over all prime

ideals of

f i n i t e l y many

v

The quadratic

symbol

(b)

in

to

2

power 0v

and

and the product

Ordv(b) / 0

of

F).

extends

0nly

and each such

v

will

a .

residue

and

b

(the ring of integers

will be such that

be relatively prime

has a solution

OF

by

-i

symbol

(a)

otherwise.

is then

1

The quadratic

if

x 2 =a

power residue

is ordv(b) v/S

42 Now consider the congruence

subgroup

rl (~) = {[ ~ bd] { SL(2,0):

(2,20)

a

~

d

~

1,

o

~

O(N)].

Clearly

(2.21) where

rz(N)

: a F n a ~ I
GO ~ =

~ Gv, 0 the product of the connected components of vES 9 ~ a b the a r c h z m e d e a n completions of Go Note that for any [c d ] in SL(2,0),

d

is relatively prime to

Proposition

(2.22) If

(-~)s : (-~)

y = [a b]

(2.23)

2.16.

x(y)

belongs =

If

n

VE S to

d

(c,d)

c

and

2o

is relatively prime

V

FI(N) , put

((~) ife/0 s

otherwise.

to

c

and

2, let

25 Then

%(Y)

(2,24) for

x(~)

=

~ rl(~)Proof.

From the definition of everything in sight it follows

tha t

s~(~) =

n Sv(~) = v

n

sv(Y)

v finite c = 0

Thus the Proposition is obvious if

(since both sides

of (2.24) are then one). Assume now that relatively prime.

c / 0

and recall that

c

and

d

are

By Lemma 2~ #c,d, det(y)) v

if

c / 0

and

c ~ U

v

Sv(Y) = t~

1

otherwise

Therefore

sv(~)

= I(~

aet Y)v if vfc otherwise

and consequently s&(y) =v~c(C,d(dety ))v" By assumption, det(Y) det(y) m I(4).

is a unit for each

v.

Moreover,

Thus by Proposition 2.1, and the product fromula

for Hilbert's symbol

vINc(c'det Y)v = v~c (c'det(Y)) v~SH (c, d e t y )

= I.

Consequently v ~c (c,d(det y) =v~c (c,d)v, which means it suffices to prove

(2.25)

v~c(C'~)v = (~)s-

26

To simplify

matters,

(The Proposition To prove

(2.25)

Chapter

13):

c

2,

and

assume

is undoubtedly recall

F

the following integer

(~)

(w,d)v

=

~

class number

true without

for each prime

(2.26)

has

basic w

one.

this assumption.)

formulas

in

F

([14],

dividing

both

w

divides

YES

v]2 (Supplementary c

but not

Reciprocity

Law)~

for each prime

which

2,

(2.27)

(~) = (w,d) w

H

(w,d)v

vcS

v12 (Power reciprocity odd

v

dividing

rewritten

formula). c,

Note that if

(w,d)v = i.

d

is a v-unit

Consequently

(2.26)

for

can be

as

(2.28)

=

(~)

H vlc

(w,d) v

vZ2

~ ws

(w,d) v

vl 2 ~

ordw(C) Now suppose of the power for each

w]c

c :

residue

~

wLc

w

By the m u l t i p l i c a t i v i t y

symbol,

(2.28)

(u,d)v = i.)

Two special

number

(2.29)

cases

in SeCtion

field,

can be multiplied

cIC,dlvl (v (c,dlv. V~Soo

Thus the proof is complete.

interest

(2.27)

to obtain

=

implies

and

3.

(Units play no role since

[ac ~]

~ FI(N)

[] of Proposition

2.16 will be of particular

The first assume

F

in which case

X(Y) = I (~)

l

0

if

c ~ 0

otherwise

is a totally

imaginary

27 since

(c,d)~

is identically one.

Corollary 2.17. Mi!nor-Serre [2]). Character of

(Cf. Kubota [19], cf. Theorem 6.1 of BassIf

F

is totally imaginary, X(u

FI(N ), i.e.

Proof.

Since

one on

G0K ~ N0

FI(N ).

Since

in

F

: X(YI)X(Y2)

is totally imaginary, ~

G~.

YI

X(u165

So suppose

and

u

u

and

belong to

defines a

for all

u165

is identically

Y2

are arbitrary in

GF,

s~(~l~ 2) : ~(yl,u165 (by (2.18)).

But

~A(yI,u

= i.

from Proposition 2.16.

Thus the Corollary is immediate

[]

The existence of such a character (with the assumption that F

is totally imaginary) provides a starting point for Kubota's

recent investiations

([21] through [23]).

Its relation to the

congruence subgroup problem is discussed in [2]. The second example I have in mind assumes

X(') = I - ( i ~

Now

X

c

0, c > 0

if 0

c f=i 0. 0' c f( 0

no longer defines a character of

trivial extension of system" for

~I(N).

(ez+d) I/2. Then

Here

G ).

w I/2

of dlmension

1/2.

or

d>0

and

FI(N)

X(Y)

d (

(~

is not a

is still a "multiplier u = [a ~], let

is chosen so that

C

J*(y,z) =

-~/2 < arg(w I/2) ~ ~/2.

and

X(Ylu ~I~l)~(Y2) y ~FI(N).

However

More precisely, if

IX(Y)1 : I

(2.31) for all

if

F = Q. In this case,

I.e.,

J~ (u J~(u j* (~1u ,z) X(~)

z)

is a multiplier system for

FI(N)

2.3.

Well's Metaplectic

Representation.

In this Section I want to sketch Well's metaplectic suitable

representation

and reformulate

for the c o n s t r u c t i o n

general

theory of the

parts of it in a form

of automorphic

forms on the m e t a p l e c t i c

group. Roughly associates

speaking,

a projective

tation).

This

associated produces

to each abstract representation

representation

multiplier

operates

symplectic

group

is

in

is of order two.

If

H

as a central topology.

Thus Well's

extensions

is a Hilbert

space,

subgroup of

U(H)

A projective

of

G

in

(unitary)

homomorphism

canonically

U(H)

this map from

ST2(F)

when the

to

~*

from

associated

U(H)

(2.32)

of projective

imbeds

representation

in the obvious

G

to

of

G

in

U(H)/T

way

H

(or multiplier)

represenrepresentations

a Borel c r o s s - s e c t i o n m =fo~*.

must a u t o m a t i c a l l y

If

is

9

to such projective

and introducing the map

G

representationre-

equipped with the strong operator

These are obtained by choosing

U(H)/T

its

they determine.

the torus

Our interest will be in the cocycle

~*.

of the group and

some basic properties

and the group

then a continuous

tations

L2

represen-

SL(2).

We begin by recalling representations

one

(Weil's m e t a p l e c t i c

a two fold covering group which repreduces

underlying

group,

f

of

f(T) = i,

be Borel and satisfy

r e) ::I

and

(2.33)

~(gl)~ g2 ) = ~(gl, g2)~(gl, g2)

for all

gl, g2 c G.

T

(by the associative

on

which G

with values Note

that

cross-section

~ f 9

Here

in

~

is a Borel f u n c t i o n from law in

G )

G • G

to

must be a two-cocycle

T .

above obviously However,

if

f'

depends on the choice of is another

such cross-section,

29

the cocycle it determines will be c o h o m o l o g o u s to projective

Thus each

r e p r e s e n t a t i o n u n i q u e l y determines an element of

A n y Borel map f r o m is called a m u l t i p l i e r simply,

a .

G

to

representations

any a - r e p r e s e n t a t i o n , projection from

set

U(H)

satisfying

(2.32)

representation with multiplier

an a - r e p r e s e n t a t i o n )

projective

U(H)

to

and a

H2(G,T). (2.33)

(o~

and all such r e p r e s e n t a t i o n arise f r o m

as above.

More precisely,

~* = po~,

where

U(H)/T

.

Then

p

~*

each p r o j e c t i v e

representation

representations

with c o h o m o l o g o u s multipliers.

~

is

is the n a t u r a l

~ : f~*,

lles in the c o h o m o l o g y class d e t e r m i n e d by

if

9

and

In this sense,

is e s s e n t i a l l y a f a m i l y of m u l t i p l i e r

T h r o u g h o u t this paper we shall want to d i s t i n g u i s h b e t w e e n representations

of the m e t a p l e t i c

and those that do not.

Moreover,

those that don't w i t h cocycle In general, group

G

~enuine

if

~

~(t)

= tol

we shall often want to confuse

r e p r e s e n t a t i o n s of

T

P r o p o s i t i o n 2.18. section

g(g)

= (g,l)

r e p r e s e n t a t i o n of (a) of

The map

~. ~og

GL(2).

of the torus we shall call

for all

t e T ~ G 9

For the sake of

e x p o s i t i o n we assume below that m u l t i p l i c a t i o n by some fixed cocycle

Z2

is an extension of the locally compact

by some subgroup if

group w h i c h factor through

in

~

is d e t e r m i n e d

a . Let

g : G~

and suppose

denote the (Borel) ~

cross-

is a genuine

Then: from G toU(H)

is an a - r e p r e s e n t a t i o n

G; (b)

The c o r r e s p o n d e n c e

~ ~ ~og

is a b i j e c t i o n b e t w e e n

the c o l l e c t i o n of genuine r e p r e s e n t a t i o n s tions of (c)

of

~

and ~ - r e p r e s e n t a -

G; This c o r r e s p o n d e n c e p r e s e r v e s u n i t a r y e q u i v a l e n c e and

direct sums. Proof.

Part

(a) follows i m m e d i a t e l y from the fact that

3O

g(glg2 ) = e(gl, g2)t(gl)g(g2).

To prove

e-representation

and define

of

G

in

H

~(t~(g)) Then

~

and obviously

~

"genuine".

into

~i

and

is an

~

U(H),

whence

The rest of the proposition

follows from the fact that a n operator e-representations

~'

= t~,(g).

is a Borel homomorphism of

continuous,

(b), suppose

A

intertwines the

if and only if it intertwines the

corresponding ordinary representations

~I

and

~2"

We shall now describe Weil's construction in earnest. The symplectic

groups of Weil's general theory are attached to

locally compact abe!ian groups

G

so the notation

denote the value of the character at

x e G.

isomorphic

x*

in

G*

<x,x*>

(the dual group to G)

Since Weil's theory is essentially empty unless to

G*

Example 2.19.

G

is

we shall assume throughout that this is the case. Let

denote a local field of characteristic

and V a finite-dimensional non-degenerate

will

vector space defined over F. Fix q to be a

quadratic form on

additive character of

F

zero

V

described

and

T the canonical non-trivial

in [46].

<X,Y> = T(q(X,Y)),

The identity

(X,Y c V)

where

q(X,Y)

=

q(X+Y)-q(X)-q(Y),

establishes a self-duality for the additive group of equipped with its obvious topology. applicable

in particular

to

G = V

V

Thus Well's theory will be together with

(q,T).

This

example in fact will suffice for the applications we have in mind. For each

w = (u,u*)

the unitary operator in

u'(w)~(x)

~n L2(G)

G x G*

let

defin@d by

= <x,u*>~x~u).

U'(w)

denote

31

Then

U'(Wl)U'(w2) = F(Wl, W2)U'(Wl+W2)

where

(Wl, W 2) = if

W i = (Ui, U[).

That is,

U'

of

G • G*

with multiplier

I/F

U(w,t) comprises

(w e G ~ G, t e T) law given by

(wl, tl)(w2, t 2) = (Wl+W2, F(Wl,W2)tlt2).

In particular, extension G

and the family of operators

a group with composition

(2.38)

of

= tU'(w)

representation

is a multiplier

the multiplier

of

G x G*

by

and denoted by

exponentiation

A(G) o

0

i

x*

0

0

i

T

[U(w,t)]

The crucial fact is that in fact, U(w,t)

U'

G = ~, A(G)

group

is (the

group

A(G);

A(G)

may be viewed (2.34)

is an irreducible

is the unique irreducible

or as the

fixed.

Therefore

representation;

representation

of

at least the first

part of the result below is plausible. Theorem 2.20 automorphisms the normalizer

of of

(Segal). A(G)

Let

B0(G )

leaving

~

in

T

denote the group of

pointwise

L2(G),

fixed,

and let

Bo(a) be the natural projection. In other words,

an

(in which case it is denoted by ](G)).

U(w,t)

pointwise

determines

the Heisenberg

equipped with the group law

group of operators

T

Yn case

is the center of

G • G* • T

which leaves

which is cal~ed

of) the familiar Heisenberg

In general, either as

T

representation

Then

PO

is onto with kernel

there exists a multiplier

T.

representation

A(G)

32

of

B0(G)

in

L2(G)

whose range (choosing the constant in all

possible ways) coincides with

BO--~-~.

formula

defines an irreducible unitary

uS(w,t) = U((w,t))s)

representation of

A(G)

equivalent to

Therefore

U.

which fixed

some unitary operator a scalar in

Indeed if

A(G)

U((w,t)s)

r(s)

in

s c B0(G),

pointwise and hence is

: r-l(s)U(w,t)r(s)

L2(G)

(uniquely determined up to

r(s)

determines a multlp]!er representation of representation)

satisfying

P0(r(s))

B0(G )

(the Well

: So

This "abstract" Weil representation is relevant to B0(G)

is the semi-direct product of

abstract symplectic group which reduces to More precisely,

s~nce

s

in

B0(G )

of the form

(w,l) ~ (w~,f(w)).

form

where (w,t)s

a

Thus on

= (w,t)(a,f)

is an automorphism of

(GX G*)* SL2(F)

fixes

completely determined by its restriction to

(~,f)

for

T) and s ~

since

the

SL(2) and an

when

G = F.

(l,t), it is

G• G* • [i]

G • G* • T

where it is

it is of the

= (wa,f(w)t),

G • G*, f~ G • G* ~ T

is continuous,

and

f(w1+w2) F(Wl,W21 : f(wl)f(w7

F(Wl~,W2~)

(2.35) Conversely,

each pair

automorphism of

(o,f)

A(G)

fixing

satisfying (2.35) defines an T

pointwise,

so

B0(G):[(~,f)] ,

with group law

(~,f)(~,~,) : (~,,f") if

f"(w)

: f(w)f'(w~).

Now let

Sp(G)

automorphlsms of

denote the abstract symplectic group of

G ~ G*

which leave invariant the bicharacter

33

F(Wl,%) <Xl,X~> F(~2,~l) = ~ Using

(2.35)

one checks

(~i = (xi'x~))

that if

(a,f) = s e Bo(G)

then

a r Sp(G).

Our claim was that the exact sequence

(2.36)

1 + (GX G*)* + B0(G) ~ Sp(G) + 1

actually

splits.

To check this it is convenient m a t r i x form.

Since each

to describe

in

Sp(G)

is of the form

~ e A u t ( G x G*)

(x,x*) ~ (x,x~)(y ~) ~:G ~ G, t3:G ~ G*, u

where

+ G*, we s h a l l ,

following

Well,

write o=

[u

,

and define ~I = [ _ ~

where

~*

denotes

~ A u t ( G • G*)

-

the a p p r o p r i a t e

is symplectic

a bd ] = a ~ Sp(G), [c

Given

],

map dual to

~.

Then

iff a~ I = I. define

fq

on

G • G*

by

f~ (u, u~) = < 2 - 1 u ~ y 6 ~, u~> . Here we are assuming, automorphism

of

G.

as we do throughout, Then

f~

and

a

that

satisfy

x ~ 2x (2.35)

is an

and the map

-, (q, f) is a m o n o m o r p h i s m In short,

(2.37) and

Bo(G )

of

sp(G)

Theorem

into

B0(G )

2.20 produces

an exact sequence

1 ~ T-~ BO--(-C7, B0(G) + 1 contains

a copy of

which splits

Sp(G).

(2.36).

Example 2.21. in Example 2.19. Sp(G)

Suppose

Then

F

SL2(F)

is local and

obviously satisfies

(Since

F

to act on

~* = a

a~ i = i; note that

in

representation attached to

L2(V) (q~V)

via

~ e F, ~ = [a c ~]eSL(2,F)

Sp(G) = SL2(F)

one can associate to each quadratic form SL(F)

V

for each

but in all other cases properly contains it.)

representation of

is as

can be imbedded homomorphically in

by allowing each element of

scalar multiplication.

G = (V,q,T)

if

G = V

From this it follows that

(q~V)

a natural projective

which we shall call the Well and denote by

rq.

Before further analyzing this representation we need to adapt Weil'sgeneraltheorytothespecialcontext Recall that

V

is a vector space over

additive character of to the linear dual by

[X,X*].

V*

F

of Examples 2.19 and 2.21. F

and

~

is a non-trlvlal

(fixed once and for all). we shall denote its value at

The natural isomorphism between

V

and

If

X*

belongs

X

in

V

V*

is then

[X,Y] = q(X,Y). Using

[.,.]

in place of

the theory Just sketched, (a)

B(Zl, Z2) = [XI,X~]

(b)

the Heisenberg group

A(V)

(c)

the symplectic group

Sp(V)

in place of

(d)

the pseudo-symplectic

group

Ps(V)

a e Aut(VXV*)

(2.38)

Ps(V)

belongs to

in place of

is of the form Sp(V)

and

A(G); Sp(G); and

in place of (~,f)

f : V • V* ~ F

B0(G).

where satisfies

f ( z l + z 2 ) - f ( Z l ) - f ( z 2) = B(Zl~,Z2~)-B(zl, z2).

This relation, of course,

is the linearlization of (2.35):

of both sides of it yields (2.35). Ps(V)

in place of the

F(WI,W2);

A typical element of

f

one can now "linearlze"

introducing:

the bilinear form

blcharacter

<-,.>

is the subgroup of

quadratic;

B0(G)

Note that

taking

Ps(V) ~B0(G).

consisting of pairs

(o,f)

In fact with

cf. (2.39) below.

Note now that (2.38) associates to each and only one quadratic form on

VxV*.

o

in

Sp(V)

~We are assuming

one F

35

has characteristic to two.)

zero, in particular,

characteristic not equal

Therefore the homomorphism

(a,f) § f is an isomorphism between

Ps(V)

Since the same map from isomorphic to

(G•

and

B0(G)

Sp(V) to

Sp(G)

the analogy between the exponentiated

and unexponentiated

~heories breaks down here.

(2.39)

(~,f)

does imbed

Ps(V)

obtain an extension of

However,

the map

~ > (~,~~

homomorphically

1

has kernel

Ps(V)

> T

in

Bo(G)

so we can still

by pulling back (2.37) through

> Mp(V)

~ > Sp(V)

~:

> 1

I The group Mp(v) ={(s,~) ~ P s ( V ) x ~ ) :

p0(~) =~(s)]

is Well's general metaplectic group. of

sp(v)

by

It is a central extension

T .

Weil's results concerning the non-triviality of

Mp(V)

include

the following: (a) Z2,

Mp(V)

always reduces to an extension of

i.e. the cohomology class it determines in

order

Sp(V)

H2(G,T)

by is of

2 ; (b)

the extension

particular,

if

F~@,

Mp(V) and

a non-trivial cover of

V

Sp(V)

is in general non-trivial; is one-dimensional, by

Z2 .

These results yield a (topological) (2.4o)

l~

z2 ~ % 7 ( v )

Mp(V)

central extension

~ sp(v) + i

in is always

36

which by Proposition

2.3 must

coincide

with

1 ~ Z 2 ~ ~r2(F ) ~ S ~ ( F ) when

V=F.

plectic

I.e.,

cover of

Although explicit

Well's

metaplectic

group generalizes

SL2(F )

constructed

in 2.2.

Well's

factor

construction

set for

S-L2(F)

as a group of operators. for this group. explicit

These

of

F

whose

basis

XI,...,X n

= ~i x , Ti(Y)

with

If

F

of

Let

on

F,

is non-archimedean~

Consequently

over

F

and in

transform

T

is the canonical

fix an orthogonal

X :~xiXi, Ti

then

denote

and let normalized

character

q(X) =q(xl,...,Xn)

the character

diY

denote

the

as above.

the limit

= me~lim~pm~i(y2)diY

(by Well

[AT]

it is an eighth

root of unity).

the invariant

v(q,~) is well-defined. equal to

forms

(q,V),

if

i = l,...,n,

Y(Ti)

this group

y

Given

q(Xi'Xi)" F,

realize

such that the Fourier

that

such that

Haar measure

is known to exist

F

OF .

V

~i=5

= T(giY)

corresponding

of

on

Recall

is

yield an

a host of representations

to quadratic

= ~f(y)~(2xy)d

(x) =f(-x). conductor

it provides

the meta-

as follows.

dTy

~(x)

immediately

it does a priori

correspond

Fix a Haar measure

(f)

does not

In fact,

terms are realized

satisfies

~ 1

exp(88

If

F=R, sgn(%i) )

if

F =C,

set

Y(q,T)

on

L2(V)

is defined

=

and

n ~(~i )

i=l

~i(x)

and define

identically to be

= exp(~

equal

r

set

Y(q,T)

as above.

to i .

The Fourier

y(~i ) Finally, transform

37 $(X) = ~ %(Y)T(q(X,Y))dY V n

where

dY=

Z diY 9 i=l

Theorem of

F

2.22.

defined

Suppose

by

we use

For each Tt(x)

(q,V,T)

Then a cross-section provided

t eF x

= ~(tx)

S-p(V)

denote

the character

y(q,r

in

)

Sp(V)

SL2(F)

by

y(q,t).

(and

(c.f.

B0(V)).

(2.40))

is

by the maps

1 b r( 1 b ) ~ ( X ) [0 1 ] + [0 1 ]

(2.42)

[1

0 -i

(~ e L2(V)).

=

r( 0 -i

0] ~

i1

0 ])~(x)

More precisely,

to a multiplier

for each

representation

of

(X))~(X)

Tt(bq

= ~(q't)-l~(-X) these maps

t e F x,

SL2(F)

L2(V)

in

extend

whose associated

is of order two.

Remark respect

SL2(F)

over

(2.4l)

cocyele

Tt

and denote

to imbed

of

let

to

2.23. Tt

In (2.42)

the Fourier

and Haar measure A

transform

dtY = Itln/2dy.

(X) = ~ 9(Y)~t(q(X,Y))Itln/2dy

is taken with Thus

9

v Proof

of Theorem 2.22. Without loss of generality we assume n t :I. Since d Y : H d.Y it is easy to check that the operators t:l ~ in question are tensor products of the operators in L2(F) corresponding

to

~i'

namely

r([ol ~])f(•

= ~i(b~2)f~(xl,

and

r([ ~ -IO]If(x) =~(~)-l?(-x) i = l,...,n. q(x) = x 2,

Thus the Theorem precisely

We note that

Theorem

SL2(F)

is reduced

,

to the case

I.A.I of Sha!ika

is generated

V:F,

[35].

by elements

of the form

38

I b [0 1 ]

0 -I [I 0 ]

and

follow Shalika relations

subject to certain

Thus one could

directly and prove Theorem 2.22 by checking

are preserved

In any case,

relations.

the resulting m u l t i p l i e r

will be denoted

rq

to the quadratic

form

representation

r

and called q.

Corollary 2.24.

r

of

SL2(F)

representation

is an explicit

attached

realization

of the

2.21).

is ordinary

q

0 -I r([l 0 ])"

and

representation

"the" Well

(This

of Example

q

i b r([0 1 ])

by the operators

that these

if and only if

V

is even

dimensional. We conclude of operators

this Section with some remarks.

B0(G)

representations decomposition

is irreducible

rq

are never

forms and group

For the case when [45],

[36],

q and

form of a division algebra already

remarked

q

importance

their for the theory

is the norm form of a quadratic [18];

for the case when

in four variables

is a quadratic

q

field,

is the norm

see [37] and

no such complete

theory of the two-fold

In these Notes we shall describe one and three variables forms on

the m e t a p l e c t i c Subsection

In fact,

the

[18].

As

results have

form in an odd number of

This is because until recently no one seriously attacked

the r e p r e s e n t a t i o n

automorphic

T(G)),

the group

representations.

in the Section I,

yet been obtained when variables.

irreducible.

is now known to be of great

of automorphic

see [35],

(it contains

Although

2.4.

group.

decomposes

GL(1)

and

covering groups of

SL2(F).

how a We~l representation

and how its decomposition GL(2)

to automorphic

The general p h i l o s o p h y

relates

forms on

is explained

in

in

2.4. A p h i l o s o p h y

for Well's

The purpose which

of this Subsection

(though unproven)

Roughly

representation.

speaking,

the idea is that quadratic

b e t w e e n automorphic

and automorphic

forms

F

representation q.

V.

The resulting

SL2(F ).

F, and Let

This group acts on

H

L2(V)

Now fix

representation

F

to be local.

and

~ ~ S-L2(F)

r

H

in

L2(V)

the operator that

Suppose

Then the p r i m a r y constituents

pr~ary

constituents

A.

its n a t u r a l action on m a y be assumed

A(h).

is so.

of

group

From T h e o r e m 2.22 it follows

commuting algebras.

zero,

the corresponding

q

denote the orthogonal

of

In fact it seems plausible

each others

group

group.

through

to be u n i t a r y and we denote it by

rq(~).

forms index I-i

forms on the m e t a p l e c t i c

on the orthogonal

space over

of

of

h c H

of these Notes.

denote a local or global field of characteristic

(q,V) a quadratic

for

a simple principle

underlies most of the results

correspondences

Let

is to describe

A(h) rq

commutes

and

A

rq

In particular,

with

generate

for the m o m e n t

of

that

that this

correspond

i-I

the commuting

to

diagram

L2 (V)

SL 2

H

leads to a c o r r e s p o n d e n c e H

which occur in

which occur in D

rq.

for "duality".

forms on

H

A

b e t w e e n irreducible

and irreducible

representations

representations

F o l l o w i n g R. Howe, D

should pair together

which occur in

A

with automorphic

rq.

This is the c o r r e s p o n d e n c e

Since the existence rest on the hypothesis commuting algebras,

r

q

and

A

some further remarks

automorphic

forms on alluded

of this correspondence

that

SL 2

we call this correspondence

Globally,

which occur in

of

of

generate

D

SL 2

to above.

seems to each others

are in order.

The precise

40

formulation of this hypothesis, reductive pairs",

in the greater generality of "dual

was communicated to me by Howe;

as such,

it is

but one facet of his inspiring theory of "oscillator representations" for the metaplectic context of

group

(manuscript in preparation).

SL 2, at least over the reals,

suspected by S. Rallis and G. Schiffmann hand,

important

literature.

special examples of

D

In the

this fact had also been (cf. [52]).

On the other

had already appeared in the

In [36] Shalika and Tanaka discovered

that the Well

representation attached to the norm form of a quadratic F

yielded a correspondence between forms on

seems to have been the first published

SL(2)

example of

and D.

extension of S0(2).

This

(Actually the

correspondence of Shalika-Tanaka was not i-i since they dealt with S0(2)

instead of 0(2).)

For quadratic forms given by the norm forms

of quaternion algebras over

F

the resulting corresponding between

automorphic forms on the quaternion algebra and by Jacquet-Langlands Shimizu.

GL(2)

was developed

([18],Chapter III) following earlier work of

In both cases,

the fact that r

q

and A generate each others

commuting algebras was not established apriori;

rather it appeared as

a consequence of the complete decomposition of

rq.

One purpose of these Notes is to describe the duality correspondences belonging to two forms in an odd number of variables, namely 2 2--T gl(x) = x 2, and q3(xl, x2,x3) = X l - X l - X 3. The inspiration for our discussion derives directly from general ideas of Howe's, an initial suggestion of Langland's, [

], and Niwa [

T~eorem 6.3, above.

].

and earlier works of Kubota

Our results,

[

], Shintani

specifally Cor.4.18, Cor.~.20,

and

lend further evidence to the general principle asserted

However, as in earlier works,

each others commuting algebras, of D.

the fact that r q and A generate appears as a consequence ofthe existence

2.5

Extending Well's Well's

representation

construction

representation

rq

it is advantageous that Well's

of

to

GL 2

of the m e t a p l e c t i c SL 2 .

group produces

There are several

to extend this c o n s t r u c t i o n

representation

for

SL 2

often

character

natural

depends

analogue of

rq

for

In this Subsection representation an analogue when

q

depends

of

rq

GL 2

to

GL 2 .

on

for

T

q

By contrast,

the

~ .

only on

More precisely,

GL 2

which

One is

not only on

q .

I want to explain how Weil's

original

i want to define

is independent

is the norm form of a quadratic

discussed

reasons now why

depends

but also on the choice of additive

a multiplier

of

~

or quaternionic

The case space

is

I shall treat the cases

in [18].

ql(x) : x 2 and

2

q3.xl, x2,x3)( following

: xI -

x~- x~

a suggestion of Carrier's.

Throughout

this p a r a g r a p h

the following

conventions

will be in

force:

l)

F

2)

n=l

will be a local field of characteristic

rn 4)

or

3

according

will denote

[[a b],l]

notation

for

for emphasis, r

or

q =ql

rn

or

by

a b [c d ] "

given in T h e o r e m 2.22 depend on

I shall denote

will be reserved

q3 ;

rq3 ;

will be abbreviated

Note that the formulas Therefore,

rql

as

zero;

rn

by

rn(~ t) ;

for the r e p r e s e n t a t i o n

Tt "

the

eventually

n

introduced

for

GL 2 .

Proposition

2.27.

(Cf.

[18] Lemma

1.%.).

If

a ~ F x,

and

a

= [s,(] ~ST2(F), 2.6).

Then

define

~a

to be

is ,~ v(a,s)]

(cf.

Proposition

42

rn(Ta~(Y) = rn(T)(Fa) for all

a cF x

Proof. and

~

and

~

SL2(F ) .

We may assume without loss of generality that

-SL2(F). -

is a generator of

Suppose first that

n =I

s=w=[[

Then 0

2a : ~ a

:

a

o ~ {1,(a,~)]

a

[-a-lo]

:

[o a - z ]

and

= (a,a) V(qn, T)(rn(T)([O a a-o l

rn(T)(~a)~(X)

])~)(x)

Some tedious computations

with Hilbert symbols also show that -I a 0 i a i a -I -1 a [0 a -1] = W[o I ] ~ [0 1 ] ~ [0 I ][l,(a,a)]

Consequently

(2.~3)

rn(~)([a.

O1])%(X) = (a,a)

lal n/2 ~(qn'Ta)~ ~(aX)

0 a-

Y~qn 'Tj

and

rn(T)(~a)~(X But

laln/2y(qn, Ta)~(aX)

rn(Ta)(~),

iff

proof. a cF x. in

The analogous

The representation

rn(T )

Suppose

rn(T)(~a ) :

identity for

[]

rn(Ta)

is independent

extends to a representation of rn(Ta)

Then for each

L2(F n)

Therefore

is completely straightforward.

Corollary 2.28. a eF x

y(qn, Ta)~(ax).

: rn(Ta~)%(X).

as was to be shown.

~ [[0i bl ] ,I]

of

) = [al n/2

a eF x

is equivalent to

rn(T )

such that

s e SL2(F).

for all

there is a unitary operator

rn(T a)(~) = R a rn(T)Ral for all

G-L2(F).

Equivalently,

by Proposition 2.27,

Ra

Ol 0],I]. i

43

rn(T)(~a) = R a rn(Ta)R[l But

G-L2(F)

Thus

rn(T)

defining

is the semi-direct product of

Conversely,

if

rn(T )

rn

i 0 FX=[[ 0 a] ].

G-L2(F)

G-T2(F),

I 0 rn(T)([O a],l)

will []

a ~ (FX) 2,

and

by

Ra

rn(~a)

If

rl(Ta)

to be

extends to

with

Example 2.29. intertwines

and

can be extended to a representation of

rn(T ) ( [0i 0a ] , i)

intertwine

ST2(F)

rl(T);

say

a = 2,

then

in particular,

Ra~(X)=I~Ii/2h(~x)

rl(~)

extends

to a representation of G-L22(F) : Jig,c] ~ G-L2(F) : det(g) ~ (FX) 2] In general, rq(T)

rq(Ta)

will not be equivalent to

rq(T).

Indeed

will extend only to a representation of the subgroup [[g,~] ~ G-~2(F) : det(g)

fixes

To get around this problem we "fatten up" extend it.

rq(~)]

rq(~)

before attempting to

This way there is more room in the representation

space

for intertwining operators to act.

Definition 2.30. Haar measure on

F

to

of the representations

Let

dt

Fx .

Let

rq(T t)

Note that the space of

denote the restriction of additive r

q

denote the direct integral

with respect to

dt .

is isomorphic to L2(F n• FX). q Moreover, the methods of Corollary 2.28 imply that r q extends to a representation of G---L2(F) satisfying

(2.44)

rq([ol 0a])r

Proposition 2.31 9

r

= laI-i/2~ta_l(X)

The action of

rq

in

by the formulas (2.45)

rq([ 0I b1 ] )~(X,t) = rt(bq(X))~(X,t)

L2(F m• F x)

is given

44 A

r (w)@(X,t) q

(2.46)

= ~(q,t)9(X,t) : ~(q,t)~

~(Y,t)Tt(q(X,Y)dtY Fn

rq( [a0 a-O 1 ] ) ~ ( X , t ) =

(2.87)

a,a)laln/2

~~(q'Tat)

~(aX, t)

and

r q ( [ o1 0a] ) ~ ( X , t )

(2. ~8) Proof.

Apply

Proposition for

GL 2

2.31

of

r

and my Definition Note

the definition implies

using formulas

construction

2.30

simply

and (2.48).

In fact thi{

to me by Cartier

reformulates

~(q~,ta) = y(ql, t )

all

iff

g ~ GL2(F )

ral

0 r l ( [ 0 a a])

awhile

ago

his ideas.

-1/2~ (

aX, ta -

2)

commutes w i t h

det(g) c (FX) 2

This

. rl(g)

is consistent

for

with

2. 13(a).

Now it is natural

to ask what

in the present

is that the orthogonal of similitudes Suppose is

(2.46),

directly

rq

that

checks t h a t

2.Z~ takes

[]

integral.

we could have defined

(2.45),

Thus one e a s i l y

Corollary

of direct

was communicated

q

a o rl([ o a])r

(2.49)

= la -1/2%(y~,ta-l)

of

group

of

Roughly q

speaking,

is simply

of Subsection

the answer

replaced

by the group

q .

first

FX=GLI(F)

context?

shape the philosophy

that

q :ql

and its action

" in

The group

of similitudes

L 2 ( F x F x)

of

ql

is given by

(A(y)~) (x, t) = l al - 1 / 2 9 ( a x , ta -2) Note that

A(y)

clearly

Now suppose symmetric

matrices

q =q3"

commutes If

F3

with

rI .

is realized

with coefficients

in

F

as the space of

then

q(X) :det(X).

2x2

45

The group of similitudes precisely,

of

q3

is essentially

GL2(F ).

More

define go X = tg X g

when

g e GL2(F)

action of

and

GL2(F )

X c F 3.

in

Then

q(g~X) = (det g)2 q (x).

L2(F 3 • F x)

The

is given by

(A(g)~)(X,t) = I det gl 2 ~ ( g , X , t ( d e t g)-2) and

A

once again commutes with

The result now is that into irreducible representations GL2(F)

rI

(resp.

representations of

which occur

on the metaplectic

GLI(F) in

r3 .

of

(reap.

A ).

GL 2

A ).

5.5.

representations

GL I

rI

(resp.

(reap.

r3 )

automorphic

The global results

obtained or expected are described The local analysis

indexed by irreducible

irreducible

group which occur In

which occur in

should decompose

Globally this means automorphic

be indexed by automorphic forms on on

~

r3 )

forms

should forms

that can be

in Subsections 6.1 and 6.2.

is carried out in Subsections

of

4.3, 4.4, and

2.6,

Theta-functions Classically,

series

attached

lated in SL 2

a.utomorphic to quadratic

representation

forms forms.

This

theoretic

from theta-

is the procedure

terms by Well

reformu-

in [47].

For

the idea is this. Suppose

ratic

F

form in

is a number n

field and

variables. e(~)

Piecing of

are constructed

together

ST2(~ )

v

of integers

of

Fv

As in Subsection of matrices

place

2.1,

integer

Proposition

GL2(0v)

~(r

s

attached

rv(q )

to

define

0(rq(~)~)

= 0(~)

Fn

For

GL2,

F

let

0v

by

rq

is

for all

q.

Uv

place

of

its group

representation K vN

with

of units. of GL2(Fv)

is the subgroup a ~ I

divisible

by

and

s e SL2(F).

the point

of

denote

Let in

of

4.

Thus

the ring

rv(q)

denote

L2(F n x Fx).

GL2(0v)

c ~ 0 (mod N).

consisting Here

K~ = GL2(O v)

N

if

is Fv

characteristic. 2.32.

Suppose

is class

q = ql

I, i.e.,

or

q3"

Then for almost

the restriction

has at least one fixed vector. v,

on

a representation

that

~(rq(S)~

~

quad-

2.32 below.

and Weil

[~ bd]

has odd residual

v,

:

a non-archimedean

the corresponding

every

a distributuon

In particular,

is Proposition

a positive

Define

is defined with the property

This is the theta function

For

is an F-rational

local Well representations

SL2(F)-invariant.

departure

q

~v0 e L2(F n x Fx)

of

rv(q)

More precisely,

to

for each

by n

O .... Xn, t ) ~v(Xi,

(2.50)

Here

i0

denotes

=

(

~ i=l

the characteristic

the characteristic

) | v

1U N

function

v denotes

10

v of

OF

and

IuN v

function

of

Ur~v = { y e U v ' y i 1 (mod N ) ] .

47 For all odd

v,

(2.51) for

rv(q)(k)~o = go

k e K N. V

Proof.

The group

GL2(0v)

[oi b1 ]

(b

is generated by the matrices

e ~

W

and

i 0

(a

[o a ]

~ Uv).

Thus it suffices to check (2.51) for these generators. Recall that our canonical additive character 0v.

In particular,

t e U v.

~v(tbq(X)) = i

Therefore

if

rv(q)(l b)~~

the Fourier transform of

i0

q(X) e O v, = ~v~176

is

i0 .

V

Thus

o o rv(q)([ 01 a])~v(X,t) = lal-i/2~~

obvious since

a

To define

G~,

b e 0v, and Note also that

rv(q)(w)~~

) = ~v~

ta -I)

=

oix ' t)

is

U v.

rq

globally,

and to introduce theta-functions

we need first to define an appropriate • ~.

has conductor

V

V

The fact that

e

T

on

~

We shall say that

if

~(X,t) = N ~v(Xv, tv)

~

on

~

on

space of Schwartz-functions • ~

is "Sehwartz-Bruhat"

and:

V

(i)

~v(Xv, tv)

creasing on (ii)

is infinitely differentiable x

• Fv

for each archimedean

for each finite

v,

~v

and rapidly de-

v~

is the restriction to

of a locally constant compactly supported function on (iii)

for almost every finite v, ~v = ~o

~v x ~v

F~v+l;

(the function defined

V

by (2.5O)). Denote this space of functions by A ( ~ is dense in in

L2(~

L2(~ • F~)

• F~),

• F~).

Since ~ ( ~

we can define a representation

through the formula

rq

• FI) of

48

V

By virtue of Proposition least all

% e ~(~

2.32 this definition

x F~).

is meaningful

Indeed for almost all

v,

for at

~v e GL2(Ov)

O

and

%v = @v"

~(~

X F~).

rv(q)(~v)%v

By continuity,

The role of character

Thus

~v T

of

on

~

For each

Proposition

(unitarily)

in

to

L 2.

is now played by a non-trivial

corresponding ~ e ~(~

2.33.

Z

to

• F~)

q (or rq)

define

i. e. ,

8(~,g)

are on

.

y e GL2(F),

8(%,Yg) = @(~,~); e (~,~)

Proof. y

(rq(~))@({,D)

For each

(2.52)

with

operates

again belongs

by

8(~'g) :

with

and rq(~)%

F~.

defined as follows.

and

rq(~)

in the local theory

The theta-functions

a~

= %v

is

GL2(F )

We may assume without loss of generality that

is a generator of b s F.

T = H~

rq(~)%({,~)

invariant.

GL2(F)o

Suppose first that

~ = [I,i]

i b Y = [0 1 ]

Then

V

rq(y)~(X,t)

= T(btq(X)@(X,t)

trivial on

F.

= ~({,9)

Now suppose it follows that

for

y : wo

But

q

is F-rational.

({,~) c F n • F x

and (2.52) is immediate.

From Well's theory

y(q, Tt) -" 1.

Moreover,

(cf. Theorem 5 of [47])

Itl : i

if

Therefore, r~ rq(W)~(g,n)

= :

~1~1n'/2 ~

Thus

y(q, Tn)$(~g,~ )

$(~,~)

t e F x.

49

= z ~(g,~).

The last step above

i aO] Y = [0

with

= lal - l / 2

~(~,~a-1)o

But

rq(y)%({,9)

= Z 9({,q)

Observation 2.34. automorphic

form on

Nevertheless, zero.

odd function

a e F x.

in

Xo

formula.

By formula

lal -1/2 = 1.

(2oI$8)

Therefore

as d e s i r e d .

The f u n c t i o n

~,

i.e.,

for example,

For simplicity, that

summation

8(9,g)

8(~,g)

always defines

is always

it may often be the case that

Suppose,

follows

from Poisson's

suppose

Finally rq(~)~(y,~)

follows

Then

suppose

that rq(~)~

also that

rq([ -i0 - ~])~(X,t)

But by the GF-invariance

of

8(~)

[ o

invariant.

is identically

= -@(X,t),

i.e.

~

is also odd for all

~ e ~.

F = Q.

(2o~7)

From formula

= -~(X,t),

-1

e(~,

~(-X,t)

GF

an

is an

it

i e.

o

]7)

: - e(~,~).

8,

-i ~]~) Therefore

8(~,~)

We close the basic

i 0~

this paragraph

by demonstrating

how

8(~,~)

generalizes

theta function 2~in2z n ~

Proposition = n ~p

Itl -l/q

where

e- I t ! 2 ~ x 2

2~176 ~p = ~o P Then

Fix

n = i, F = ~, and

for all finite

if

8(%,g)

g = [[yl/2 0 = yl/As(x+iy)o

p

rq = r I.

and ~ (x,t) =

xy-l/2 -1/2 ]' 1 ], y

Suppose

50

Proof.

Since

[yl/2 xy-1/2 0 y-I/2 ] =

Z rq(~)%({,~). - .

I x

[yl/2

[0 1 ] 0

lyll/4

- =

o

y -I/2]'

e-2W7~ 2 e2~i~2x~.

({,~) Here we choose 0p

for

only if

T = ~ Tp

p < ~. { e ~

Thus and

with

({,~)

T (X) = e 2~ix contributes

9 = i (recall

and

Tp

trivial

to the summation

~2 = 102~176

l.e.,

e2~in ~ ( x + l y ) as was to be shown~ Note that for

a c (0,~),

a o r([ 0 a ] ) ~ ( ~ , ~ )

Thus

8(~,~)

is actually

and

=

-I/2

as above,

=

ai

=

al-i/2

=

tl-i/~ e-ltI2~x2

defined 7

@

z~

on

~(ax, ta -2)

Ita-21-1/4

e-ltl2~x2

above

on

w

AutomorPhic Forms on the Metaplectic Group: Global Theory.

3.$o Preliminaries. We start by describing the correspondence between classical forms of half-integral weight and automorphic forms on the metaplectic group of Section 2.2. For convenience, we shall assume that the ground field is Thus we shall consider functions in

f(z), defined and holomorphic

Jim(z) > 0], satisfying

(3.1)

f(yz)

for all N

Q.

~ e El(N).

=

f ~#az+b~ j = Y(~)(cz+d)k/2f(z)

Here, as before,

k

is an odd positive integer,

is a positive integer divisible by 4, and

plier system of dimension

1/2

We shall also assume that of

FI(N ).

cusps, i.e., [38]. )

is the multi-

described by (2.30). f(z)

is holomorphic at the cusps

In fact we shall assume that f(z)

X(?)

is a cusp form.

f(z)

vanishes at these

(For precise definitions,

see

Suffice it to say that cusp forms have Fourier expansions of

the type oo

(3.2)

with

f(z) =

Z a(n)exp(2wi n z) n=O

a(0) = 0, and for these forms,

(3.3)

If(z)IIm(z) k/$ < M~

Nhat we want to show is that such forms correspond to particularly nice functions on by

~

~

The space of these forms will be denoted

S(k/2, rl(N)). Recall that

K

denotes the maximal compact subgroup of

whose connected component is K* = S0(2)

=

Jr(8)

~cos@ - sin@ = Lsin@ cos@ ]]

GL2(~)

52

0 ~ ~ < 2~.

with

is still abelian. realize with

--# K

r(e)

.~

Although

To describe

as

~/4~ ~

~s above.

K*

with

0 ~ e ( 4w.

K-~

must make

elements ~*

its character

rather

[~(e)

=

~sine

of pairs

to

[[r(e),~}]

cose~

The isomorphism

~(0),-i}

Z2

between

correspond

to

these

7(2~)

(and at most

can have this property). on

than as the group

it is convenient

rCOSe - s i n e ] ] =

are of order two

non trivial

group

Thus

-~ (3.#)

S0(2),

does not split over

realizations

since both these

one non-trivial

Consequently

of

element

each character

of

of --# K

must be of the form k

(3.5) with

~k/2(~(@)) k Next

: e

odd. recall

the Casimir

operator

for

S--~2(~)

in terms

of the

local p a r a m e t e r i z a t i o n

2 = (x,y, e).

Here

~ =

yl/2 [ o

The Casimir

x y-z/2 y-l/2]

operator

,

Proposition to a function

(1)

on

~(yg) : ~(g)

homomorphically (2)

3.1.

~f

~({~)

in

y > O,

is the differential

2. 8 2 ~2 ~ = -y ( - - ~ + T )

(3.6)

on

T(e)

S--L2(~)

for

for

0 ~

e < &~.

operator

~2 ~x~---e

Each cusp form

SL2(~)

= {~(~)

- y

x ~ ~, and

f

in

S(k/2,FI(N))

corresponds

such that:

y ~ SL2(Q) ( r e c a l l

y ~ { y,S~(y)}

via the map ~ ~ Z 2,

SL2(Q )

i.e.,

~

imbeds ;

is a genuine

function

G--L2(~);

(3)

~(~k0) = ~(~)

(4)

~(g ~(9))

for all

ko

= ~k/2(T(6))~(g)

in

~o n SL2(~);

for all

T(B)

in

K-~ ,

i.e.,

53

_

transforms under (5)

_

w

K*~

according to the character

viewed as a function of

satisfies the differential

S-L2(~) alone,

ek/2; ~

is smooth and

equation

A~ = - ~k (k~ - 1)~; (6)

~

is square integrable;

\,~

I,~(~)1 2 ~

SL2(Q) (7)

~

more precisely,

< 0%

2 (~)/Z 2

is "cuspidal" on

NQ~ ~ ~(E~I

S-~2(~), i.e.

l]~)dx = 0

(This expression is meaningful

(for

a.e.

since

N~

g)

lifts as a subgroup of

'~2 (~).) The significance

of (6) and (7) ~s that

the space of square-integrable The significance

~f

will belong to

cusp forms for the metaplectic

of (5) is that

-k/~( ~k -

i)

group.

will be the

eigenvalue of the Casimir operator for the representation of

S--L2(~) with lowest weight vector

k/2

(~k/2

in the notation

of Section ~)o To define

~f

and establish Proposition 3.1

we use a few

Lemmas. Lemma 3.2.

Suppose

= ~k

with

y =

-~2(~)

~y~S~(y)]

in

~ = (g,~)

belongs to

S-~2(~).

Then

o

SL2(Q ),

k 0 =[k0,1 ]

determined up to left multiplication

in 4 n by

SL2(~), and g~ in

[Y0, S~(Y0)]

in

r l ( N ) = SL2(Q) n SL2(R).~~ 9 Proof.

By "strong approximation"

SL2(Q)SL2(~)~ ~.

More precisely,

for

SL(2),

g = yg k 0 ,

with

SL2(~) = g~

in

SL2(~)

54

determined up to left multiplication by elements of [g,C] = {y,S~(~)} [g~,r

r

with

r

:

~&(Y,g~)~&(Yg~,k0)S&(Y)"

FI(N ).

Thus

{ko, l] If

yO:[Yo, S&(YO)]

r

straightforward but tedious computation shows that [g,{] = {YYo I, S~(yY01)]

(~og~,r

=

For

[y0g~,{'][ko, l},

{Yo,S~(Y0)~{g~,r

g : [ac db ]

in

with

[]

does not define a factor of automorphy for choose

w I/2

so that

~ = y

in

(We agreed ~o

Indeed by the

(Hecke [i~], pp. 919-940),

x(~)(e~+d)I/2

FI(N ) .

Now we can define

and fo~

FI(N ).

-v/2 < arg(w I/2) ~_ ~/2.)

functional equation for the theta-function

for

J*(g,z) = (cz+d)i/2

SL2(~), recall that

f(z)

(3.7)

in

~f~

For

S(k/2,~l(~)),

g-- = (g~,{)

in

ST2(~ ),

set

set

~f(g) = f(g~(i))j*(g~,~)-k.

Note that in (3.7), g~(z)

if

g~ = ([ca bd ]'{)"

(3.8)

az+b : c--Yg~

Thus (3.1) rewrites

itself as

f ( V ( z ) ) = j* (-{, ~ ) k f ( ~ )

for all

-{ -- { ~ , S ~ ( y ) ]

in

rl(~).

(By P r o p o s i t i o n

2.16,

S~(u

= x(Y):) Lemma 3.3. S-~2 (~), i.e.

(a)

J*(~,z)

defines a factor of automorphy on

55

(3.9) for all

g,g'

in

SL2(~); (b)

a character of Proof.

K*~

(a)

The restriction of non-trivial on

takes its values in F

Z2, namely

to

K~

defines

~k/2"

The function

F(g,g,) =

Since

J*(g,i) k

j~(g g,,z) j~(g,g,(z))J(g,,z)

Z2

and is obviously a Borel map on

SL2(~)xSL2(~).

can also be shown to satisfy the factor set relations

and (2.2) (cf. Maass

[24], pp. 115-116),

J*

automorphy on the extension of

SL2(~ )

computations

is the cocycle

then show that

F

(2.1)

defines a factor of

determined by F. Further ~

described in

Theorem 2.2. (b)

the

By (a), and the fact that ~(e)

restriction

of

(j.)k

--g.~ K But by definition,

follows

to

-'g" K

stabilizes

clearly

defines

J*((l,C),i) k = ~k = C.

from the('definition

of

i e Jim(z) >0],

a character

That

of

J* = ~i/2

~1/2" ~

Proof of Proposition 3.1. Note first that

~f(~)

is well defined even though

(3.7) is determined only modulo

TI(N).

g-~ in

Indeed by (3.8) and (3.9),

j~ (~o,~(i)kf (~(i))j~ (yo,~(i))-hj~ (~, k)-k = for each

YO

~f(7) in

Properties

TI(N ). (i) - (4) are trivial.

analyticity assumption for tion (applied to

f(z)

To prove (5), recall our

Then use straightforward

~f(x,y, 0) = yk/4f(x+iy)e-i(k/2)6)

computa-

to verify that

56

ik ~-- (9

ik

A
~+

i

(~-~ + i ~7)f(~)

Y

ik

- I[(I[kk _ l ' y k / % f ' z ' - [) ) e "-~ (9 To prove

(6), note by (1)-(4)

function on the homogeneous

I~f(g)

that

l2

actually defines

a

space

ss2(r kZZ2(~) / Ko which by Lemma 3.2 is isomorphic

to

Consequently

(3. lO)

~l~f(g) 12dg = ( c o n s t a n t ) ~ l f ( z ) i 2 y k/2 dxdy 2 Y

and the right-hand

side of (3.10)

To prove

simply compute:

(7),

is finite by (3-3).

I X --

1

--

2 ~([o 1 ]g)dx = 2 ~([o ~]g)dx : 2 f([o1 ~J2-(i))J*( ~ [o1 x1]g~,i)-kd~ Z~R

= J*(~, i)-ka(o) where

a(O)

completes

is the zeroth Fourier

the proof of Proposition

Remark.

The Petersson

coefficient

of

f .

This

3.1.

inner product

in

S(k/2;FI(N))

is

given by the integral

~f(z)g---(~)yk/2

dxdy 2 Y

F with

F

diverges

a fundamental

domain for

FI(N )

in general for non-cuspidal

It converges,

however,

for modular

in

H .

This integral

modular forms in

forms of weight

M(k/2,rl(N)).

1/2 ( k = l ) .

57

We conclude

this paragraph by remarking

with the obvious modifications, S-~2(~)o

For

~,

denotes

The result satisfy

@A

of the identity

which are trivial on

3.1 again correspond

3.1 also generalizes

in

GL2(R).

Z0

and

to hlassical

to arbitrary

number fields.

(3.11) must be replaced by the identity

G,~ =

with

on

component

k/2 .

Proposition However,

the connected

of Proposition

forms of weight

in place of

X0

is that funetions

(1)-(7)

G~

N

aA:aQa GO

is valid for

3.1,

one must appeal to the fact that

(3 11) where

that Proposition

aj ~G~.

h 0 N D G~a.G K~ j = l % g c~U

Consequently

one starts with

fj

of weight

k/2

and defines

fj

and each piece

G-ai GUK~o "

trivial on

and satisfying

correspond

Z0 ~

to collections

~

on

~

functions

~

forms

(3-7)

of Proposition

forms of weight

situation

in the case of totally imaginary fields.

metaplectic

by applying

(1)-(7)

of classical

of "quaternionic

classical

The result is that functions

only "new" feature of this general

on a product

h

can be

refer the reader to Proposition

K

3.1 now

k/2.

The

f

is defined

and the corresponding

invariant.

5 of Kubota

on

is already apparent

In this case

half-spaces"

to

[21].

For details we

3.2

Factorization Let

2.2~

~

of Automorphic

denote the homogeneous

According

to Proposition

pond to special functions functions

Forms.

on

[

Z ~ GF\[ A,

3.1, forms in

in

T2(~),

satisfying

by Jacquet-Langlands,

space

S (k/2, ~I(N))

corres-

the space of square-integrable

~({~) : {~(g)

theory for

as in Section

GL(2),

for

{ { Z 2.

Inspired

we now make the following

definition. Definition 3. (or

An

a generalized

irreducible

of

v

~v"

which occurs T

of

in the

~A

in

~2([).

forms for

GL(2))

of the local groups

G-v.

factor as "products"

of

Thus we shall first recall

theory tailor fit to the

~v o

Suppose of

weight.) is an

is to explain how such represen-

some basic facts from representation groups

~

representation

in this paragraph

(like automorphic

representations

form of half-integral

representation

of the regular

Our purpose tations

automorphic

unitary

decomposition

a.utomorphic representation of the metaplectic group

~v

is an irreducible

Suppose

also that

v

unitary "genuine"

is finite and odd.

is class i if its restriction

to

[

v

representation

Then we say that

contains

the representation

Y0: (k,~) ~ at least once. to

Kv

Equivalently,

contains

the restriction

the trivial representation

of this representation

of

Kv

As we shall see in Section 6, such representations identity

representation

exactly

v

where the product unitary,

for all

class i for almost all

the

of the form

~v ~

is over all places

and genuine

will contain

once.

To make sense out of a product |

at least once.

of

F, and

Yv

v, we must suppose that

v, say for

v ~ SO 9

is irreducible, ~v

is

59

Let

[~]

determined

denote the c o l l e c t i o n

by

[~v ]

above through P r o p o s i t i o n

denote the space of 0 ~v

vector uniquely set

in

~$).

HS

K v.

| H v. vcS i m b e d d i n g of H s

Gv Hv

v ~ S O , pick a unit

i.

0 ~v

is

For each finite

Then if in

of

2.18 and let

By the above remarks,

equal to

( | ~). v vcS'-S

and

For each

up to a scalar of modulus

isometric

lim HS,

space

(or

fixed by

determined

is a natural ~ ~ |

~v

Hv

S ~ S O , set

of ~ v - r e p r e s e n t a t i o n

S' ~ S, there

Hs, , namely

we can define the p r e - H i l b e r t

Consequently

(by completion)

a Hilbert

s p a c e H = ~ Hv

on

S which the

~A-representation

V

acts as follows:

~ = | ~v

is any decomposable

element

of

H,

V

0 ~v = ~v

with

If

for almost

every

v, then

~'(g)~ = | ~v'(gv)~v for

g - (gv)

in

G~.

By the c o n s t r u c t i o n ~-representation (by Schur's determined

of

Lemma).

above we obtain from

G~

The u n i t a r y

"genuine"

T=|

~ " (even though V

product larly,

product

say that

~

an irreducible

unitary

is factorizable

~A

of the representations

makes sense

of c o r r e s p o n d i n g m u l t i p l i e r

given

of

We shall denote it by

as the "tensor product

this

representation

V

V

~

a unitary

which can easily be shown to be irreducible

by it is also irreducible.

and refer to

[~v ]

o n l y when v i e w e d a s a

representations

of

representation

Y

if it can be realized

as

Gv).

of | ~

Simi-

~

we in

v

this sense. Theorem 3 . 5 . of

~

unitary

Every irreducible

is factorizable.

representations

I.e.,

~v

of

unitary

genuine

representation

there exists a family of irreducible

~v'

completely

determined

by

~,

60

genuine

for every

in the above

i for almost

local group

this T h e o r e m we need to appeal ~v

is type i.

is primary.

That is,

Lemma 3.6. G 2.

Z2

see Howe

[iY] and H a r i s h - C h a n d r a

sentation

Suppose

E1

of

and

Gi

with

completely

~2

on

(unitary)

7i

determined

and

Gv0 x G~0, where

Lemma 3.6,

G~

v

unitary

a collection

by

GA

is a product

G

of

unitary

Then:

is of the form of

Gi

~. Using it we can sketch [ii].

as the direct product (gv0)

G~

with

gv0 = i.

determined

unitary

~'v0-representation

of irreducible

G

~i-representation

of adeles

of an irreducible

and an irreducible

of

[2~].

and write

consists

on

the spirit of [9] and

7', the ~ - r e p r e s e n t a t i o n

the tensor product

~

groups

is type I (for i = 1,2).

in Mackey

~v

G2, and that every ~i-repre-

(up to isomorphism)

fix a place

of

[15].

of locally compact

~-representation

of Theorem 3.5 following

First,

Gv0

GI

an irreducible

This Lemma is implicit the proof

is a product

which is a factor

every irreducible 71 | 72

G

of

it is:

Suppose also that the cocycle

of cocycles

to the fact that each

each factor r e p r e s e n t a t i o n

The Lemma which we shall use to exploit

and

v, such that

For a proof of this fact for representations

which are trivial on

GI

every

sense.

To prove

~v

v, and class

by

By

~,

is

~v0-representation of

Gv0 ' .

~v-representations

of

Thus we obtain of

Gv

for

each place v. Next we need to show that these for so

Kv v

for almost

v.

of adeles

kw c K w

for

of

w ~ v.

G~

contain a vector invariant

For simplicity,

runs through the sequence

the subgroup where

every

~' v

assume that

of rational primes.

of the form Clearly

~

Let

F = ~

denote

(l,...,l,kv,...,kw,...) converges

to the identity

61

as

v + ~.

a vector greater

Thus one can show that

invariant

for

K0 v

than or equal to

class i for each

when v O.

H (the space of v

is s u f f i c i e n t l y

From this it follows

large, that

say

~' v

is

v ~ v O.

Finally we need to check that normal basis

[fv, j]

in each

Kv

v ) v 0.

Since

for each

~') contains

~' = | ~'.

Hv ~'

such that ~ ~v'

and

it will suffice to construct

am isometry

patible w i t h the actions

G~.

By Lemma 3.6,

of

So choose an ortho-

V

H can be expressed

fv, l

for

are both irreducible

from

as

is invariant

~ Hv

to

H

( ~ v Hw) | H$

com-

for each v.

\

Thus for each the form KV~

( ~

~

f ~

V

VW'

l)f,

V

On the other hand,

to a scalar) each

v ~ v0,

v ~ v0

some vector with

f' V

H

we get an imbedding

invariant

a unit vector

w~v Hw = Hv ~ ( ~ v

the only vector in

f( ~ I) ) w>v fw,

in

H~ of

w i t h the required

Hv)'

for

in H'

invariant

under

(w~v Hw)

into

compatibility

is of

invariant

V

and

Kv

fv, l K v. H

is

for

(up

Thus for (namely

conditions

satisfied.

3.3

The S p e c t r u m Let

GF\GA

L2(~)

of the M e t a p l e c t i c

denote

which are

the Hilbert

trivial on

ZO

Group

space of m e a s u r a b l e

functions

and square-integrable

on

on

= ZO GF~A,

a quotient

s p a c e of f i n i t e

volume.

The m e a s u r e on

~

is such that

f(~)d~ = ~ [ Z f((g,~))~dg x ~z 2

if

Let L2(X)

L2(X)

and

for future

Set

denote the subspace of

~2(~)

decomposition

(A~

GF\a~2(~)/Z ~

x : x/z 2 :

of

its

orthoeomplement.

L2(X)

are w e l l - k n o w n

reference.

B~

For details,

see

denote the non-unimodular

is the group

complex number

of positive

s,

let

diagonal

e sH(b)

b =

denote

Z2-invariant

functions

Basic facts

concerning the

but we recall [i~

and

them here

[17].

subgroup matrices

in

~A~ in

of

A .)

B~

For each

the character

-> a2

of

a I 1/2

B~.

Here

I=-I

~(b) : log

In general,

suppose

~2 (3.12)

g = nh t akz t

with

nc N~ ' h t =

k c K= K *~

Let on

(3.13)

G~

[e 0

K v , and

H(s)

[a I 0 0 -t ], a c A~ = [ ] ~ A~:]all e 0 a2

z c Z ~0 9

Then

denote the space

=

la21

= i~,

H(g) = t.

of c o m p l e x - v a l u e d

such that

~ I~(k) l 2 dk <

measurable

functions

63

and (3.14)

for all

b ~ B'.

The induced

R(x,s)

operates

in this

G~, e sH(b))

by right

translation

space

= 0.

Moreover

basic

fact is that the direct

(3.15)

R(x,s)

Re(~

intertwines of

G~

series

and

L2(X)

for each

~

in

if for each

~(xk)

space

of such

will

on

B~\G~ If

R(x,s)

this

of

be denoted

Thus

| (T) = ~, L satisfying (3.13).

~ ~ (<) c ~, for all

are equivalent.

The

of the regular

c

one needs

then

s ~ {,

~

is contained

representation

L.

T

~ ~ C~(B~G~),_

the regular

by

when

R(~,s)dlsl

~

which

~

and is unitary

representation

to introduce

Eisenstein

~(s).

if

x [ G&,

~(g)

integral

but to prove

More precisely,

(T)

R(x,-s)

with a subrepresentation in

s+l 2

representation

= Ind(B~,

Re(s)

T

a7 = I~I

~(bg) = e(S+l)H(b)~(g)

K

(~)

is said to be of type in a subspace

is equivalent

of

to

and the collection

the Hilbert

~(g)e (s+l)H(g)

and the Eisenstein

space

L2(K) T.

on

The

of all such

of measurable

belongs series

to the space

associated

of

to

is by definition

~(g,~,s) =

z

~(~g)e (s+l)H(~g)

~BF\G F

However, defined

this for

series

Re(s) = 0

It is known that and satisfies

converges

only if

Re(s) > 0. Thus

only in the sense E(g~,s)

the functional

of analytic

is meromorphic equation

E(g,~,s)

is

continuation.

in the whole

s-plane

64

E(g,~,s) where

M(s)

with

is the"T-component"

R(x,-s).

E(g,~,s)

Moreover,

is regular

particular, Re(s)

in

of

of the space

satisfying

s = i.

E(g,~,s)

is well-behaved

on

X

Tc

(3o15)

with

then has discrete

In

when

T c.

The ortho-

spectrum,

is denoted

as follows.

the closed

the cuspidal

imply that at

of

denote

M(s)

R(x,s)

except possibly

and may be described 2 L0(X)

intertwining

Re(s) ~ 0

and may be used to intertwine

complement

Let

of the operator

the properties

the function

= 0

2 Ld(X),

= E(g,M(s)~,-s)

subspace

of functions

in

L2(X)

condition

1 ~iJg)dx ~ o ~([o

S F~

for

a.e.

for

G~.

g ~ G~. If

This

L~(E)

is the space

denotes

which are realizable

of square

the subspace

as residues

of

integrable

of functions

E(g,~,s)

at

in

s = i,

cusp forms L2(X) then

L~(x) = L0(x) | and this decomposition Before

returning

has a pole at (with

X

s = i

a character

In particular, such that sentation

suppose

~l~1(x) of

G

v

=

is compatible

with

G~o

to the metaplectic

group

only i.f the T-type

of

of

Fx~ x ~i

Ixl

induced

and

whose ~2

and let

square

~

note that

is T(k) = X(det k) 0 is trivial on F~ ) .

are quasi-characters 9(~i,~2)

E(g,~,s)

denote

of

~v

the repre-

from the character

x 1 iI1121a21 a2 of v

B v. equal

Then each constituent to the one-dimensional

of

L~(E) quotient

is of the form of some such

| ~v

with

p(~l,~2)~

65

Now we return ~

in

Z2(~).

to

This

which are genuine, crucial point Eisenstein

~A latter

i.e.

is that

series

and the natural space

satisfy ~

consists

To be more specifie~

of functions

~({g)

: {~(g)

over

Bi.

splits

and induced

representation

Thus

representations

for arbitrary

for

T ~

of in

{ [ Z 2.

L2(~) The

one can introduce

as for

G~.

g { ~,

write

are as in

(3.12).

= nht~[z --i [ ~ A~, [ c K, andn, ht,

where

function

H(g) = t

and

is again well-defined. aI x

H([

],~) = log a2

0

and

8(~) : e 2H(~) Let on

~

describes

denote

B~\~

z

such that

~(~{)

In particular,

a I 1/2 I I ~2

the modular

the Hilbert

Then the

space

: {~(g)

function

for

of measurable for

{ c Z2,

%. functions

and

K (By

Iwasawa,

sentation

of

NOW fix

[~ = [ ~ ) . ~

If

define

~ ~

~

~

and

is an irreducible ~

as before

unitary

so that

repre-

~ = | (~). T

and set

E(~,~,s) --

(3.16)

~

~

~(~)e

(s+l)H(~)"

~ BF\G F

This

is an Eisenstein

since

~

function E(~,~,s)

(3 917)

splits on

series

over

Z ~

generalizes

GF

which

on the metaplectic and it converges

is left

classical

E ( ,zk , s , )

:

z c,d

invariant

series

group 9

for Re(s

> i

for

As such,

G F.

of the type

ez+d (~)(T---:~) Icz+~1

It makes

cz+dl

~ 2 S

toa

sense

66 and

(3.~8)

E(u,O,s) :

This

last function

in our set-up K-type.

Such series

described

first

E(~,~,s) in

by Siegel

with

corresponds

defined

in

over

~(~---d) which

~

of'~rivial"

[21] and recalled

by ( 3 . ~ ) group

series

over

i.

to an Eisenstein

~.

[~2], then more

in Section

This

recently

series was by

in [39].

In general, representations let

to some

were studied

for the metaplectic

investigated Shimura

is an Eisenstein

corresponds

The function series

Z x(~) v(~u) s+l r Xr(N)

~e sH(~)

these Eisenstein

series

with a subrepresentation

denote

intertwine of

~.

certain

induced

To be more precise,

the character aI x

=

by

~.

([0

a2]'~) ~ ~eSH(~)

T h e n the induced

representation

~(~,s) : Ind(~, [~, {e sH([)) is unitary

when

Re(s) = 0

R(~,s) for all space

s e C. ~(s)

satisfies

= ~R(7,s)

(R(~,s)

consists

and

is a genuine

of functions

~

representation on

~

of

~.)

Its

sucm that

~ ( ~ ~) : ce(S+l)H(~)~(~)

for all

b e B~

,

and

S I~(k) l 2 dk < ~. Note that if

~ ~ ~,

the function

~(~)e (s+l)u(~)

belongs tZ:o

67

the space of

H(~,s)

In fact the correspondence

is an isomorphism between (3.19)

~

and

~(s)

and i~ is easy to check that

~(~)E(g,~,s) = E(~,I-l(s) ~(~,s) I(s)~,s).

Clearly (3.191) says that representation of

T.

E

intertwines

However, as for

~(~,s)

GL(2),

with a sub-

this statement is

meaningless unless the analytic continuation of

E(g,~,s)

is

exploited. Roughly speaking, the analytic continuation and functional equation of

with

E(~,@,s)

an i n v a r i a n t

subspace

poses discretely. direct sum of

makes it possible to intertwine

of

only at

L~dd(~),

will

E(~,~,s)

E(~,~s) ~(~,s)

be t h e

T2(~) satisfying --2Lo(E )

at poles at the right of Re(s) =0.

coincide with those of the operator with

R(~,-s).

In [i0] we show they occur

s = 1/2 and generate representations which are simple to

describe. to

it

decom-

(7) of Proposition 3.1, and a space

consisting of residues of

intertwining

call

the space of functions in

the cuspidal condition

M(s)

whose o r t h o c o m p l e m e n t

This orthocomplement,

T~(X),

These poles of

T2(~)

L2(X))

A new feature of the decomposition of

T2([)

(as opposed

is that in the latter only infinite-dimensional

opposed to one-dimensional) of cusp forms.

Thus if

representations occur outside the space

~ = | W--v so occurs it is natural to ask

what its local components are? cerns Eisenstein series.

(as

This last question really just con-

The remarks below are included to explain

the question and to show why Sections 4 and 5 are prerequisites for a meaningful response to it.

88

Note added (March 1976). continuation and poles of

Precise results concerning the analytic

E(~,~,s)

were announced in [I0].

proofs of these results (at least for certain special choices of appear as part of a joint work with Jacquet [54].

Complete ~) will

3-%-

Odds and Ends, Recall that

[(~,s)

from the character

is the representation of

Ce sH([)

of

Z~ = N~ AFA~Z2o

G-A

induced

Thus by inducing

in stages,

R(~,s) = I n d ( B ~ , ~ , T ( [ , s ) ) Here

T([,s)

denotes the r e p r e s e n t a t i o n of

triangular subgroup of Now let

R(~)

representation of on

~.

%)

induced from

ZA

(the f u l l

Ce sH(~)

denote the subrepresentation of the natural A--~ in

Since the quotient R(T)

which aets on genuine functions

L2(AFA~k~) AFA~ ~

is compact,

= 9 m Y

with the summation extending over 'the "genuine" dual group of --I A~ (m is a non-zero integer for only countably many Y). Y

If we view see that

T(~,s)

T(~,s)

as a representation of

is the tensor product of

dimensional representation

~e sH(a)

of

TA

R(~)

AOo

it is easy to

with the one

Thus

(3.20)

R(~,s)

In (3.20), T

denotes an irreducible unitary representation of

occuring in

R(~)

and

representation of ~(~)e sH([)

of

Now suppose

: ~ R(~,Y,s). Y

R(~,l,s)

~

extended to

TA

in the obvious way.

is an irreducible representations of

occurs discretely in

T

is a constituent

~(E).

of

denotes the principal series

induced from the representation

~A = A--~ x A I ~

~

outside the space of cusp forms.

~

which

Then

From the theory of Eisenstein series

for themetaplectic group (see [i0]) it follows that sucha.representation imbeds (as aquotient) in someR([,h,s)

w i t h O < s~1/2.

of these principal series representations end of Subsection 3.3 is obvious.

Thus the relevance

to the problem posed at the

70

To analyze

R(~,~,s)

it is necessary

"product"

of local representations

abelian.

Thus it should not seem surprising

dimension

greater

than one

and

Lo

to decompose

However,

~

~

as a

is not

that each

L

~ is infinite dimensional.

has So to make

sense out of the product

~ = |

(3, 21)

we must (as in Section 3.2) interpret representations irreducible identity

it determines.

unitary

it in terms of the cocycle

This will be possible

~v-representation

representation

of

Av~ Kv

of

Av

will contain the

exactly once.

will turn out to be four-dimensional,

since each

(We note that

at least when

v

Nv

is

finite and odd.) The facts needed to make Sections ~ and 5. character it follows

(3. 21) precise will appear in

We shall see that each

of a subgroup

B'v

%

v

of index four in

is induced from a ~vo

From this

that

R(~,Y,s) = | ~v(Xv,~l,~2) V

with

and

~i,~2

quasi-characters

series representations

of

F x. v

It is precisely principal

of this type which will be discussed

in

Sections 5 and 6. Let us return now to E(~,~,s)

to the right of

the local components

~ (E)

with

As already noted,

Re(s) > 0

all occur at

of the corresponding

will be quotients

~v(~l,~2 )

--2 L0(E ).

~l~21(x)

the poles of s = 1/2.

constituents

~

of

of the local representations = Ixl I/2. S:

~-~

7~

Moreover,

the global map

Thus

71

alluded

to in Section i will

The map

S

of

will associate

Eisenstein

series

~

non-trivial

on

S(~v)

(3.18)

S(~v)

s = i.

function

0n the other hand,

of

is a representation

Then the infinite representation S

of

G~

of

if

component

~v

~

of

representation

series on

G~.

Shimura's

E(u,s)

with respect

with respect

to

r0(N )

S(~)

is

k - i.

equation

for

for

F = Q

component

is

will be a discrete

Indeed Shimura's

k/2(k 2 3). series

Thus the map

Shimura's

theory.

will not exactly contain forms are automorphic

and such forms need to be lifted above

to a slightly larger cover. functional

for

to Hecke's

suppose the infinite

whose lowest weight

(cf. Section i).

series

eigenvalues

whose lowest weight

the correspondence

GIA

For finite

of a principal

will also play a crucial role in explaining Actually,

of

will be consistent with those

F = Q,

of

unitary

F = ~(~),

The corresponding

of

and the eigenvalues

ring. T

as follows.

generated by the

the one-dimensional

of index

the zonal spherical

When

|

will be the class i quotient

representation

of

Z 2.

to the representation

generated by the poles of Eisenstein v,

this phenomenon

will eventually be defined for irreducible

representations S

"explain"

(A four-fold

~(z)

cover will do since the

implies that for

J~(~zY2'i)

~

yi,Y2

in

r0(N ),

~(~i~2)

with

and

Cd = i

(modulo 4).

or

i

according

if

d

is congruent

to

Thus our results at best are "consistent"

i

or

3

with

Shimura's. As already suggested, more carefully

all these questions will be described

in Section 6 after the necessary

sentation theory have been obtained.

facts from repre-

w

Local Theory:

~.i.

the Archimedean

Basic Representation

theory.

We start by developing in a form suitable The theory for

the representation

for our definition

GL2(C)

Subsection.

Section

3 suggests of

G-~2(R)

representations

of the subgroup

We start by dealing with 9.

Important

our attention

which are trivial

S--L2(R)* = [(g,~)

of

G-L2(~) S(~v).

and will be recalled

that we restrict

representations

theory

of the local map

is well-known

in the next

denote by

places.

e -C-L2(R): det(g)

S-~2(R)

which

subgroups

are

to

Z0 ,

on

briefly

i.e.

to

= ii}.

(for the moment

only)

we

i ~ ,i)]

: {([o l ] = inverse a

image

of

a

0

0

a

A : [[

_i ] ]

in

-@.

0

A 0 = [[

_l ] e A: a > O] 0 a

= inverse

image

of

S0(2)

= K

and

= centralizer Recall

that

~(8)

LsinsrC~

=

M = ~i 2

K

A0

-sinS],cos~ j

with

obviously on

K.

with the group of rotations

group

imbeds R~)

in

O _< e < ~ .

is a cyclic

symbol being trivial

(~.i)

A0

may be identified

and hence

Since

of

M

is the inverse

of order

as a subgroup

the Iwasawa

~ = N A 0 [.

~

of

generated

~

decomposition

image by

(the Hilbert for

~

is

of

73 Also,

since

~

is a connected

Harish-Chandra's particular, admissible

general

semi-simple

theory is completely

every irreducible (its restriction

unitary

to

and every such representation unitary)

principal

Let

B

triangular

T

decomposes

of

G.

of

In ~

of some (possibly non-

of the following

image in

~

is

with finite multiplicities)

is a subquotient

denote the inverse

(~.2) Fix

~

applicable.

representation

series representation

subgroup

Lie group with finite center,

type.

of the upper

Then

Z : M A 0 N.

a quasi-character

of

~

M, and consider

of

AO,

an

irreducible

the representation

of

T

representation

(actually

~/N)

defined by (4.3)

~|

The resulting

a n) = ~(~)~(a)o

Banach space representation

~(~,~) : Znd(%~, ~| will be unitary

iff

unitary principal

f

on

is, so in general,

9(~,t)

is a non-

series representation~

The space of functions

~

p(Z,~), ~

call it

B(Z,~),

consists

of measurable

such that

(~.4) for

b ~ B,

and

Here

6(~)

= 8([

a

for

B.

Since

x

_1],~) = Ial 2 denotes the modular function 0 a ~ = N A o K , B(~, T ) may also be described as the

space of square integrable

functions

on

K

satisfying

74

for

~

in (the finite group)

be trivial on SL2([~ ))

iff

analyze

Z2

Bn K=M.

Note that

(and hence define an ordinary

T

is trivial on

Z 2.

M

T(yJ) j=1,2,3,4,

(and hence

T =0

or

and

I.

Before

factors

na ~NA0,

and

how

0~0<4~,

Z2

representations

= ~|

reduces

~m(g)

depending

(a)

m

is an even integer

(b)

m

is an odd integer

(c)

m = 4/2,

with

g -= 1(4)

if

~ = 1/2;

(d)

m = 4/2,

with

% ~ 3(%)

if

T = -1/2.

In any case,

the appropriate

(if at all)

we

on

~

by

ime

is an integer of h a l f - i n t e g e r

if

y2={-12,1 ]

B(h,T).

define

if

Since

if and only if

p(h,T)

basis for

~m(naT(0)) m

or-1/2.

through

describing

a convenient

(4.6)

for

(unitary)

= e~iJ T T =0, i,i/2,

p(h,T))

introduce If

Here

of

and they are given by the characters

where

must

representation

these representations

(4.5)

T

will

Thus we want to further

There are p r e c i s e l y four irreducible of

p(~, T )

T =0 T =I

on

~. More precisely,

; ;

collection

[~m ]

and

comprises

a basis

B ( ~ , ~') o

a 0

Since every q u a s i - c h a r a c t e r

of

A0

is of the form

~([

_l])=a s, 0 a

will h e n c e f o r t h k

be treated as a complex number,

will denote an odd positive Lemma 4ol.

(a)

integer.

However,

invariant

subspaces,

p(h,O)

if

s.

Also,

integer greater than or equal to three. is irreducible

~ = k - 2,

B(h,O)

= B_(k,-l) =

E

m>X+l

m~ X+l(S)

if

h

contains

namely

B_(X+I)

namely

r

is not an odd exactly two

75 and

%(~+i)

= %(k-l)

:

E

r

~i-(h+i) m~(h+l)(2) if

~ =-(k-2),

B(h,0)

contains

exactly

one invariant

subspace,

namely

B(h+l)

:

@~m'

Z

m-~ (h-l)(2) and this representation B(h,0)/B+(k-l)eB with

h = k-3 (b)

2

subspaee,

(k-l)~

k ~ I(~),

p(h,l)

if ~ / •

B(h,I/2)

are identical

A = k-2; k-2 --7-,

contains

However 9

exactly

if

one invariant

9

namely

Bl/2(h+l) =

B(h,-I/2)

Z

~%'/2

m,2k m'~k.(5)

-

and

for

replacing

is irreducible

with

to the quotient

The results

(an even integer)

p(h,•

h = k-2

is equivalent

contains

the invariant

%1/2(_h_i)

=

~

subspace

~%,/2

~

m'<-k

m'~-k@) On the other hand, the invariant B-i/2(k/2)~

B(h,i/2)

contains

subspace Similarly,

contains

B~i/2(k/2 contains

k-2 ~ = - - 7 with

if

2);

if

Bl/2(-k/2) if

and (k-2) 2

~ =-

Bl/2(-k/2

k ~ 3(4),

B(~,I/2)

B(h,-i/2) '

with

contains k = 3(4)

+ 2)

and

B(~,-I/2)

~ = - Ik-2)2 '

and

k ~ 1(4),

the invariant

subspace

~i/2, ~_ ~- ~k + 2) 9

B~/2(k/2

- 2)

contains

and

then

contains then

B(h,i/2)

B(l,-I/2)

76

(c) in

The subspaces

B(Z,•

(•

have infinite

and the representation

(in the obvious irreducible to

B~

sense)

to the theory

representation

Z = -1/2)

B(-I/2,1/2)/BI/2(3/2) Part

the Lie algebra covering,

of

is equivalent

Proof.

~

in

argument

at

used

BI/2(I/2)

to

SL2(~ )

is complementary the

(corresponding

representation

h = i/2). and is well-known.

to prove

sketch

~

-~; for example,

to the quotient

(a) concerns

first

at

(corresponding

group we simply

Consider

theory

codimension

it works

the argument

equally

Since

well for the

below.

the elements

1 i],

v+ = [i -1

1-i]

v

= [-i -1

and U= in the

eomplexification

clearly

span the

through

the

of

~.

These elements

Furthermore,

they act

on

B(X,T)

f

df = ~-~ (g exp(tX))It=0 ,

in

(hence also for

B(h,~) X

and arbitrary

Thus one obtains

complexified

Lie algebra

a representation

(hence also

in the space of K-finite

For each basis (4.7)

element

X

in the Lie

in the complexification

extension).

algebra)

Lie algebra

formula

for smooth

algebra

of the

complexification.

p(X)f(g) valid

[ 0 1 -1 0 ]

of

p(X)

the universal

functions

in

by the obvious of the

enveloping

B(~,~).

B(h,~),

p(U)~ m = im~ m

and

(~.8) These

~(v~)~

formulas

m =

are independent

(~+l~m)~m•

of

T.

2 9

Moreover,

the first

implies

77

each invariant contains.

subspace of

Therefore,

if and only if from (~.8).

B(h,~)

since

p(X)

is spanned by the

p(h,T)

is (topologically)

is algebraically

irreducible~

The following representations

representatives

for the equivalence

representations

of

p(it,~)

(b)

The representations

p(s,O)

the representations

p(s,~I/2)

The representations

BI/2(~)_ , B~I/2(-k/2),

in

~

in

exhaust a set of

B1/2(1/2)_

and

for

t ~ ~ -i < s < !, s ~ O;

-1/2 < s < 1/2, s ~ O;

B~(~(k-l)),

6+~i/2(-k/2), and

The r e s u l t s

results for

with with

with

The trivial representation

Proof.

B~(T(k-2)),

B-i/2(k/2)_ of

SL2(~),

with

k >_ 3~

and the representa-

B~1/2(-1/2).

SL2(~ )

a r e a g a i n well-known and t h e

S-~2(R) can be deduced from Sally ([33] and [3~]).

In all cases except B(X,T)

the Lemma follows

G:

The representations

tions of

irreducible

classes of irreducible unitary

(a)

(d)

it

[]

Lemma 4.2.

(c)

~m

(a) one must introduce a new inner product in

(or an appropriate

we must recall that

subspace thereof).

p(O,l)

Also, to be precise,

is reducible and the direct sum of

two irreducible representations.

[]

Terminology. The representations series representations, I

-

p(~,T)

in (a) will be called continuous

those in (b) complementary

series representa-

-

The only equivalences

tions.

The representations representations (or k/2), ~k-i

(or

of

K

in

+ ~k-i

are

p(~,T) = p(-h,T).

in (c) will be called discrete series

of lowest

(or highest)

weight

etc., whichever is appropriate. ~k-l' +

or

e

They will be denoted by

~k/2,... ) to emphasize that among the characters --

appearing in them, the lowest is is

k-i (or -(k-l))

-i(k-l) 8

, the lowest in

e i(k-l)9 --~k/2

is

(The highest e

ik/2

,

78

and so on.)

A l l these

and all but

m2

The

and

representations

lowest weight

1/2

character of ~ i _ set*of S-L2(~).• Now we return GL2(~ )

representations are square-integrable • ~3/2 are actually integrable.

of

and

is non-tempered to the group

[g e GL2(•):

from a normal setting

subgroup.

G/N,

and

N

theory

We recall

group a

is a normal G.

If

h

__

ml/2"

ml/2

-1/2.

The

image

To describe

to (a special

has

in

its representat~n

case of) Mackey's

of representations

induced

this theory here in an abstract

since we shall use it again

Suppose compact

of Clifford's

the inverse

= ~i)~

to appeal

.-J:

-G,

but does not vanish on the elliptic

G* ,

det(g)

theory it is convenient generalization

in (d) will be denoted --+ ml/2 has highest weight

on

in another

subgroup

in Section 5.

of index two in the locally

is a non-trivial

is a representation of

context

coset representative

N, then

a

of

is said to be self-

conjugate if it is equivalent to its "conjugate" representation ~h(n) From Mackey

= ~(n h) = ~(hnh-l).

[26] one can deduce

Lemma 4.3.

Suppose

unitary and irreducible. (a)

If

~

and equivalent (b)

If

two irreducible with

H

are as above,

and

~

is

Then Ind(N,H,~)

is irreduclble

Ind(N,H, h ) ; is equivalent

to

representations

~h,

whose

Ind(N,H,~) restrictions

is the sum of to

N

coincide

~; (c)

N

and

is not self-conjugate, to

~

N

the following:

All irreducible

as above

(assuming

To apply this The crucial

unitary

everything

Lemma to

--

G*

representations in sight we fix:

facts are then included

Such representations

are called

of

O

arise

from

is type i). h

equal

to

([0i - o ],i).

in:

"trash" by P. Sally and N. Wallach.

79

Lemma 4.%.

(a)

are the continuous, conjugate of (b)

complementary,

is

representations

M-type:

SL2(~) the

Wk ; of

SL2(~)

is self-

in fact, each is conjugate to its obvious mate of opposite

for example,

p( it, 1/2)

is conjugate to Proof.

For ( b ) ,

of

and trivial representations;

None of the genuine representations

conjugate;

~3/2

v~

The self-conjugate

is conjugate to

p( it, -1/2),

~3/2' and so on.

Part (a) is proved in Section 2.1 of Gelbart [8].

start

with the identity a

(4.9)

x

h([

a

x

_l],~)h -I = ([ Oa

_l],sgn(a)~). 0a

(This follows from Lemma 2.11 and the definition of the Hilbert symbol for Clearly

~.)

fh

Now for each

is zero iff

f

f

in

B(k,I/2),

put

fh(~) = f(hg).

is, and

fh([ [) = f(h [ g) = f([hh g) 81/2(Y)~| But by (4.9), p(X,i/2)

k|

with

n) = k|

p(k,-i/2).

to invariant subspaces, Proposition 4.5. G*

). SO the map

f ~ fh

intertwines

Since it also takes inyariant subspaces

the proof is complete.

[]

Every irreducible unitary representation

of

belongs to one of the following series of representations: (a)

the continuous p(~r

where

B* = [[

aI

x

0

a2

G* = [g ~ GL2(~):

k~([al 0

series of representations

= Ind(B*,G*,~) ]]

is the full triangular

det~g) = ~i],

~ ~ JR,

~

subgroup of

and

x ]) = lalik[sgn(al)]C[sgn(a2)] ~ a2

~ = 0

or

1, and

80

the complementary series

(b)

0

and

X

is between

the discrete series representations Vk_ I = Ind (G, G*, ~k_ + l) where

Vk_ 2,

(d) or

where

1;

(c)

and

p(hO, n)

k ~ 3

:

Ind(G'G*'q-1)

is odd;

the one-dimensional representations

g § [det(g)] c,

r

=

0

1.

Proposition 4.6. G*, non-trivial on

Every irreducible unitary representation of

Z 2,

belongs to one of the following series of

representations: (a)

the continuous series representations p(h)

where

:

Ind(Z,G-~,X|

:

Ind(Z,~-~,X|

)

~ s i ~+X ; (b)

the complementary series representations

p(h) where

o < ~ < i/2~ (c)

The discrete series representations

= Ind(~,G*,Vk/2) (d)

~i/2 = Ind(G,G ,Vl/2)

the representation

Proof of Proposition 4.5.

Observe that

Ind(~, a*, ~(~, c) ) : Ind(a, a*,ind(B, G, ~c) ) = Ind(B,G*,~ c) : Ind(B*,~*,Ind(B,~*,~ where

~r

[~ x ~[ c a_l]) = lal sgn(a)] o

But

))

Ind(B,B*,~r

=

~0 9 h I r r

Thus the Proposition follows from Lemmas ~o2,~o3, and 4.4. Proof of Proposition %. 6.

[]

Apply Lemmas 4.2 through 4.4 directly.

4.2.

The Local Map. As in Subsection A.I,

group

G

[g e GL2(~):det g = •

will denote and

SL2(~), G*

~* its inverse image in

From Section 3, and the work of Shirmua, non-trivial

correspondence

the existence of a

associating representations

to genuine representations

of

the

of

~* is clearly indicated.

indicated that this correspondence

should associate

In the next we shall construct

to this correspondence

~k/2 9

in very

in completely different terms~

certain ordered pairs of quasi-characters

(~i,~2)

certain irreducible unitary representations Suppose first that ~* non-trivial on

to

(part of) an inverse

We begin by setting up a one-to-one correspondence

of

It is also

~k-i

In this paragraph we shall define such a correspondence simple terms.

SL2(~)

~ Z 2.

of

of

between ~x

and

~*~

is an irreducible unitary representation Let

~i,~2

be quasicharacters

of

~x

satisfying the following properties: (I)

~i (-I) : ~2 (-I) = l~

(2)

~I = ~21

(3)

~lu~l(x)

-

Let

~(~i,U2 )

X

on

~+; and

= Ixl s

with

Re(s) i 0, Ira(s) k 0o

denote the representation

of

~*

Induced from

the character aI x ([0 a 2]'~ ) + ~ l ( a l ) ~ 2 ( a 2 ) aI x

of

B* = [ ( [ 0

a 2] '~ ):

ala2 = •

By Proposition #.6 there will be exactly one pair satisfying of

(~i,~2)

(1)-(3) and such that some (possibly trivial)

~(~i,~2 ), call it

we can speak of

~

and

~(~i,~2 ), is equivalent to ~(~i,~2 )

Hence

in one and the same breath.

Now we want to associate to each such pair irreducible unitary representation

~.

subquotient

of

G*o

(~i,~2)

So let

an

p(~l,~2 )

82

denote the representation of aI

[ 0 of

B*.

If

G*

induced from the character

x

a 2] -~ ~l(a)~2(a2 )

%(x) = ix] ~t , then

representation

p(hO,0)

irreducible

(~I, U2)

k = hl-A2.

satisfying

representation of

itself if it is irreducible,

(1)-(3) above a

G*, namely

p(~l,~2 )

or "the" irreducible unitary representa-

tion defined in a subquotient of reducible.

is equivalent to the

of Proposition 4.5 with

Thus we can associate to each well-defined

p(~l,~2 )

p(~l,~2 )

if

p(~l,U2 )

We shall denote this representation by

The only ambiguity is when

A = I

(In which case

is

~(~i,~2 ), ~(~i,~2 )

is

either the trivial representation or the discrete series representation

~2 ).

Definition 4.~176 Let

~

denote the collection of equivalence

classes of irreducible unitary ~enuine representations If

~ e ~

W(~,~).

corresponds to

(UI,~2),

2 2) ~(~1,~2

Here

define

(respo quotient)

of

2 2 (~1,~2) is a 2 2 W(~l,~2 )

will be reducible if and only if

is reducible.

Proposition 4.8.

The map

is one-to-one from

~

represemtations

of

G*

representations

of even weight.

ont.o the collection of irreducible unitary which are either class i or discrete series

s(~k/2) and the inverse image under G*

~(HI,~2)

~(~i,~2 ), then 2 2 (respo quotient) of p(~l,~2).

is a subrepresentation 2 2) Note that p(~l,~2 ~(~i,~2 )

if

9*~

to be

is associated to the pair

according to the following convention: subrepresentation

S(~)

of

is the non-tempered

S

Moreover,

if

k ~ 3,

= ~k-1 of the trivial representation

representation

~i/2 o

of

83

Proof.

The proof is immediate from the representation

theory thus far developed.

In fact our definition of the local

map was inspired by the prerequisite that this Proposition is true. The definition of the local map for simpler since Let

G-T2(~ )

UI,~2

is the direct product of

denote quasi-characters

the representation

of

GL2(~ )

sz~d B(Pn)

of

GL2(~ ) and

with

Let

B(uI,~2)

Z2.

p(Ul,~2) ~i~2

denote the space of

the subspace which transforms according to the

unique representation of

SU2(~ )

of dimension

representation theory we shall need for two results below.

~x

is analogous but

induced from the character

of its upper triangular subgroup. p(U,u2)

F = ~

n+l.

GL2(~ )

All the

is contained in the

The appropriate references are Section 6 of [18]

and Chapter 3 of [12]. Proposition 4.9. (a)

p(~l,U2)

p >_ i, q ~ i; (b)

in this case,

If

P q Ul~21(z) ~ z ~

is irreducible if

~l~21(z) = zPz q

Bs(UI,~2 > =

p(~l,~2 ) with

p >__ i,

Bs(~l,~2 ) (e)

will denote the representation and If

~(UI,~2 )

~(Ul,~2);

q >_ i,

Z B_p+2 n3p+q(2)

is the only proper invariant subspace of o(~i,~2 )

is denoted

with

B(~I,~2); of

the representation

~l~21(z) = z-P7 -q,

Bf(~l'~2)

with

= IP-ql

in this case,

GL2(~ ) in

p ~ i,

in

B/Bs(UI,~2)o q ~ i

( n ( p+q

B(~n)

n ~ p+q(2) is

the only proper

invariant

~(~1,~2 )

will

denote

~(~i,~2 )

the representation

subspace

the representation

of

B(~l,~2); in

in the quotient

in this

Bf(~l,~2 )

and

B/Bf(~I,~2).

ease,

84

(d)

~(UI,~2 )

(~i,~2) : (u{,~) (e)

If

I T ~(~i,~2 )

is equivalent to or

(~$, ~{);

Ul~l(z)

: zPz-q,

with

a pair (Vl, V2) such that V l V 2 = ~ l ~ 2 a n d (f) is a

~(ZI,~2)

GL2({ )

with

for some choice of

the continuous

(b)

the complementary

of

GL2({ )

(~i,~2),

~

representation

representations

~ = ~ | ~(UI,~2)

~(~i,~2 )

g ~ u(det(g)),

character of

of

G-~2(C)

~

a

unitary representations (~i,~2)

and

non-trivial

by Part (d) of Proposition 4.9,

Definition 4.11.

Z2

for some pair of quasi-characters

are completely determined by

to

unitary;

{x.

an irreducible unitary representation

corresponds

~i,~2

o < s < m;

denote the non-trivial

Moreover,

with

series representations

the one-dimensional

Now let

@x.

there is

~(~i,~2) =~(Vl, V2); finally,

series ~(Ul,~2)

I~12s,

unitary character of

of

q 2 I,

belongs to one of the following series of representation:

~l~1(z) :

Then

and

Each irreducible unitarizable

(a)

(c)

p 2 1

Every irredumible admissible representation

Proposition 4.10. of

if sad only if

Let of

ml

on Z2.

UI,~ 2

and

~2

7. T

denote the collection of irreducible

GL2(@ )

non-trivial

as above, define

S(7)

on

Z2.

to be

If

7 ~ T 2 2 ~(~l,U2).

Note that the correspondence

takes continuous representations

series representations

to continuous

and class i representations

series

to class I representations.

It also takes "the" complementary

series representation

(i.e.

and

~(Ul, m2) ,

with

trivial representatiom. representation

at

Ul~ 2 = i

~iZ21(z)

at

: Izl 2)

to the

(In this sense, the complementary

s = 1/2

H

attempts to be a non-tempered

s = 1/2

II

series repre-

85

sentation

of the covering group;

cf~ Kubota

[22],

[23],

and Section 6

of this paper. Note,

finally,

that

Imdeed the only possible series representation

(L-function)

should be unitary whenever

exception

of index

at least in the comtext global

S(~)

is when

1/2 ( s ( i.

of cusp forms,

considerations.

7

~

is.

is a complementary This possibility,

should be elimimated

by

4.3

Weil's Representation in 3-varlableso Let

V

matrices.

denote the 3-dimensional We shall identify

V

space of

2x2

real symmetric

with the Lie algebra of

SL2(R)

via the map

Ex

3 Xl-X2J < - )

(Xl'X2'X3) <--)

Thus a natural basis for

61--

and

0 I1

V

-i 0 ]'

X = (Xl,X2,X3) :

-Xl+X3] : Xo

f

l+X3

-x 2

J

is provided by the matrices i 62 = [0

0 -1 ]'

0 1 63 : [i 0 ]

and

3 Z xi% i . i=l

Our purpose in this Subsection is to decompose the Weil representation attached to the quadratic form on

q(X) : d e t

(X),

defined by

equivalently, q(xl,x2,x3)

Call this representation the map

V

~ ~ w

= Xl 2

rq 9

because

rq

and the orthogonal group of To be more specific, sentation attached to

x22 - x32o

The decomposition

of

defines a representation q(X)

fix

(q,V,T)

is

(essentially)

m(x) = ewix

rq

ties in with

of

Z'E2(~)

SL2(B).

Then the Weil repre-

operates in

L2(V,dX)

through the

operators 1 b ~• rq([ o 1 ]) ~(x) : e

(4.10)

%(X)

and (%.11)

o -l -~i/4 rq([ 1 O])%(X) : e %(-X)o

By Corollary 2.2%, the representation (4. II) is not an ordinary representation ~-representation

with

~

cohomologous

factor set described by (2.6).

Thus

of

(in r

q

determined by (4.10) and SL2(R).

Rather it is a

Z2(SL2([),Z2))

to the

defines a representation

87

of

S-L2(~) non-trivial on

Z2 .

Now we introduce the orthogonal group of the proper Lorentz group in 3-variables usually denoted by

S0(1,2).

q 9

This group is

(I time, 2 space variables)

It operates on

L2(V)

as a group of

unitary operators

and obviously commutes with (4.10) and (A. II). rq

r

since it commutes with the generators

q

For this reason one expects to be able to decompose

according to the decomposition of the representation The group

SL2(~ )

H(g).

comes into play as usual through its adjoint

action Ad(g)X = gXg -I on

V.

This action preserves

provides a surjection of S0(1,2)

of

SL2(~ )

(with kernel [~K2]).

with a subgroup of SL2(~)

in

S0(1,2);

L2(V),

~(x)

q(X)

and the homomorphism

onto the connected component of

Thus

SL2(~)/(~K 2)

can be identified

the resulting unitary representation

given by

~ ~(g)~(x)

= ~(Ad(g-1)x)

will be called the regular representation of

SL2(R )

in

Our first task is to describe the decomposition of irreducible representations. representation of

g+Ad(g)

SL2(~ )

Roughly speaking,

occurs in

H(g),

a multiple of some irreducible contituent of irreducible representation of

S-L2(R).

L2(V). H(g)

into

if some primary

its space will realize rq,

hence index an

The resulting correspondence

will then be of some interest. In what follows we shall restrict our attention to representations of the discrete series and simply make parenthetical remarks

88

for the remaining representations. subrepresentation

of

mk-l" m

H(g)

Note that

constituents,

H(g)

Thus our task is to describe a

equivalent

is trivial on

including

~! k-l"

Under the action of

to infinitely many copies of [•

hence its possible

must also be.

SL2(E),

the space

V

decomposes

into

orbits of the following type: (i)

the origin

(2)

the forward

(3)

(of no interest); (reap.

[x:q(X)

= O,x 1 k 0 (resp.

a sheet

of

the

H+ = [X: the

hyperboloid Hi

The action of is isomorphic

= iX:

light cone

x 1 s 0)]

hyperboloid q(X)

m

(a)

backward)

of

two s h e e t s

= m2} ;

of one sheet q(X)

= -m 2}

9

SL2(R) to

on each of these orbits is transitive: (2) + Sg2(~)/N , H I is isomorphic to Sg2(R)/K, and

H~ ~ SL2(R)/A Our interest will primarily be in that discrete Let in

H-

H-

denote the region

is of the form

mh

f(X)dX

H-

SL2(E)

dh

with

dm

the is

to decompose

= ~

Each

X = (Xl, X2,X 3)

m = - I q ( X ) II/2 c (-~,0)

The corresponding

and

integral formula

is

0 ~ f ( m , h ) l m l 2 d m dh

invariant

Lebesgue

L2(H -) To decompose the

[X: q(X) ~ 0].

H~ - ~

denotes

and

since it is in this orbit

series appear.

xI x9 x3 h = (-~- , -~- , -~- ) s H~ .

where

H~

measure

on

on

H~

inherited

(-~,0).

= L2((-~,0),lml2dm)|

H(g) - i n v a r i a n t

L2(H~)

measure

subspace

from

Thus

L2(H[) L2(H -)

we need f i r s t

89

Henceforth, Let

Gk

k

is an odd integer greater than or equal to three.

denote the space of functions

(i)

[ ] g ~ O,

(ii) g

where

g(X)

~2

~2

[ ] = ~x12

~x22

is homogeneous of degree

k~3

on

if

k ~ 3(4);

(iv) g(X)

~x32 ;

i.e.

g

to

Gk

Thus

Gk

sentation of

if

k~l(4)

t>0 and

; ~ :I

H-

are completely determined by their restriction is a Hilbert space with inner product given by

the inner product in That

r =0

for all

and

vanishes outside

Note that such H~ 9

where

such that

~2

k-3 g(tX) =g(txl, tx2,tx 3) :t 2 g(X) (iii) g(-X) =(-i) r g(X)

V

L2(H~)

realizes an essentially irreducible unitary repreSL2(~)

can be deduced from recent work of Strichartz

([44]) and Ehrenpreis ([53])Proposition 4.12. the representation

The precise result is:

The natural action of

~k-I ~ k - I

SL2(~)

in

Gk

realizes

"

This follows from a careful study of the K-types appearing in [44] (pulled back from If we let

S0(1,2)

L2(k,H -)

to

SL2(R)).

demote the subspace of functions in

L2(V)

which can be approximated by linear combinations of functions of the form a(x) with

g e Gk

and

f(m) e L2((-~,0),

Corollary 4.13. for

H(g)

and (as an _

many copies of

~ f(-lq(x)ll/2)g(X)

L2(k,H -)

Imlk-ldm),

~k_l~k_l

9

we have:

is an invariant subspace of

SL2(~ ) -module) +

,

L2(V)

is equivalent to infinitely

9O

Now using a series of non-trivial Lemmas we shall establish the following:

for

Theorem 4.14.

L2(k,H -)

~

S-L2(R) -module)

and (as an

copies of

(a)

L2(V)

is equivalent to infinitely many

~k/2 "

Lemma 4.15. L2(k,H -)

is an invariant subspace of

with

Suppose f(m) ~ ( ~ )

rq([o1 ~])a(x)

:

G(X) =f(-lq(X) II/2)q(x)

belongs to

(the Schwartz space of

~ ).

Then:

f,(-rq(X)ll/2)g(x)

where f'(m) = e~ibm2f(m];

(b)

rq([ 0l ol])a(x) : e -~ i/4y( -[q(X)[ 1/2) g(x)

where

k-2 ~(m-lt) 2 cos ~ [ ~ + r

f(m) = e~ir

Jk_2(2mt)f(-t)t

0 Proof.

Part (a) is completely straightforward.

the other hand, (Theorem I).

dt .

5

results from formulas appearing

Roughly speaking,

Strichartz'

Part (b), on

in Strichartz

[43]

theorem describes the

Fourier transform of a function on

V

natural action of

SL2(R)

according to a given irreducible

representation of

G.

weight vector that

(~• r

k-i

on

V )

which transforms

If this representation

condition

has a highest or lowest

(3) in the definition of

on p.509 of [43] must be odd.

(under the

Gk

implies

Thus the formulas there

simplify considerably and an application of them to our situation proves the lemma.

[]

Cprollar[ 4.16. the restriction of

L2(k,H -)

rq

Furthermore,

to L2(k,H -) is equivalent to infinitely q many copies of the ~-representation of SL2(R) generated by the operators

r

is invariant for

91

[0

: f-~f'

and [01 -01] : f-~e-~i/4 ~ .

To complete the proof of Theorem %.14, observe that L2((-~,0),Imlk-ldlml)

is isomorphic to

L2((0,~),mdm)

via the

map k-i

f(r)

-~ f ( - r )

l rl 2

Thus Theorem 4.14 follows from Lemma 4.17 below coupled with some straightforward manipulations.

(A useful identity here is the

following:

~-r k-1

cos 2L 2 +.c] =e

_ _~2 ( k - l )

~i~/2

e

.)

~k/2

(resp.

~--

Lemma 4.17. is realizable in (a)

The representation L2((0,~),mdm)

the operator corresponding to

1 [o

(resp.

~]

S--~2(R)

is

the operator corresponding to

Proof.

( resp.

e ik~/4 )

Explicit realizations of the representations of

are to be found in Pukansky [32] and Sally [33]. corresponds to

R•

in [33] with

follows from Lemma 0.1.5 of [33]. Corollary 4.18.

S--L2(~)

In particular,

h =k/~.

Thus our Lemma

[]

There is a natural correspondence (the "duality

correspondence") which associates to the representation G*

of

e-~ibm2f(M)) ;

f(m) ~ e -IkT/% ~ Jk_2(2mt)f(t) t dt 0 2

~k/2

~k/2 )

so that

f(m) ~ e~ibm2f(m) (b)

--3 c

the genuine representation

~--k/2 of

~*.

correspondence inverts (part of) the local map

~k-I

of

In particular, this SR

of Section 4.2.

92

Proof. from r

q

rq

Let

r*

denote

q

the representation

( v i e w e d as a r e p r e s e n t a t i o n

depends on

~,

r* q

~k-l"

Its restriction

indexes the primary

when induced up to

contains

--g G

r* q

(Cf.

Lemma

will contain

continuous

series

The required Fourier the formula The result

is that

+ ~k-I @ ~ k - I

is

component

of

r

correspondence

which

q

~v~u/2" [ ]

to class i

as well since these representations

of

L2(V)

transform formula

~(~i,~2 )

speculations

will appear

(albeit

continuously).

is now more complicated

than

the representation

is mapped to

about how this

r3

The representation discussed

denote the restriction representation to

-~ 1/2 1/2~ ~i '~2 J"

correspondence

More inverts

in Section 6.

Concluding Remark.

isomorphic

G

in Lemma 4.15 but again it can be derived from [43].

comprehensive

the trivial

this

representations

also occur in the decomposition

to

infinitely many copies of

We note that one can extend

r'q

Although

does not.

In particular,

and this representation

let

induced

~k/2 = Ind(~'G--g'~/2)"

So now consider

S: ~ + ~

--V G

G = SL2(N)).

it can be shown that

4.3 and the proof of Lemma 4.%.) copies of

of

of

~xr3(Tt)dt.

of

of

r* q

in Section 2.5. r*q

.~-~2(~) _ ~

to in

above is essentially More precisely,

S-T21~)_ tensored with L2((O,~)).

Then

r'q

is

4.4

The basic Well representation The Well correspondence

(Joint with P. Sally)

discussed in Corollary 4.18 pairs

together discrete series representations

of

GL2(~ )

Z

with genuine discrete series representations

on

Z O.

trivial on

of

G-L2(~)

Its domain actually extends to a larger set.

irreducible

"tempered" representations

Equivalently,

of

GL2(~ )

consider all possible constituents

representation

of

GL2(~ )

in

L2(~3 • ~x).

to above pairs together constituents with constituents

of

Consider all

trivial on

GL---2(~) r I.

The correspondence

of

of this adjoint representation

modulo

(but

relating a subset of the dual group of

to the dual group of

~-L2(~).

alluded

r 3.

correspondence

Actually,

~I.

of the "adjoint"

Our purpose in this Subsection is to describe a similar simpler)

trivial

GLI(~ ) .

for convenience,

Thus we deal with

In place of

r3

we deal with

we deal with the representation G--L2(~) modulo its center and

rl(~

GLI(~ )

(0,~).

Fix

T(x) = e wix.

Then the action of

rl(m )

in

L2(~)

is

g i v e n by t h e o p e r a t o r s I b ~ibx 2 rl(m)([ 0 l])~(x ) = e ~(x) and rl(m)(Q)~(x ) : e ~i/4 ~(x).

Theorem 4.19. (a)

With the notation of Subsection 4.1,

rl(~) : ~ 1 2 R e c a l l that

sentation

~i/2

| ~s

"

is a subquotient of the principal series repre-

p(i/2,1/2)~

it has lowest weight

1/2; ~3/2

is a subi

representation (b) to

Let

of

p(i/2,-I/2);

rl(m i )

m_l(X ) = e -~ix

it has lowest weight vector

denote the Well representation

Then

-3/2.

corresponding

94

rl(T-1) = ~ / 2 Here

Y~/2

has highest weight vector

weight vector

and

--+ ~3/2

has highest

It is easy to see that the subspace of

consisting of even (resp. odd) functions is irreducibly

invariant for

rl(T+l ).

representations of

To identify the resulting irreducible

S-L2(~) one uses the models for

in Chapter IV of [33].

rI

Let

r~ (resp. r~)

m

presented

denote the restriction

to the space of even (resp. odd) functions in

irreducible representation tion

m[/2

[]

Proof (2) (Sketch). of

-1/2

3/2.

Proof (i) (Sketch). L2(~)

| ~3/2 "

r~

L2(~).

The

is intertwined with the subrepresenta-

-

~i/2

of

p(-i/2,1/2)

via the operator

~(x) ~ rl(g)~(0)~ the representation

r~

is intertwined with the subrepresentation ~3/2

d (x) = ~(rl(q)~)l x=o Proof (3) (Sketch). A basis for L2(~) is provided by the 2 functions 9m(X) = e -~x H n ( ~ x) when Hm is the Hermite polynomial of degree

m, m ~ 0.

= i(m+i/2)@ m, i.e., rE 90)

and

r~ =~i~2

r~ and

has lowest weight

has lowest weight

3/2

1/2

rl(U)9 m

(corresponding to

(corresponding to

91).

Thus

r~ : ~3/2 "

Corollary %.20. induced from

But by differentiation,

rl(T )

Let on

r~

denote the representation of

-~2(~)

~-L2(~). Then

rl = ~i/2 ~ ~3/2 " Note that

r~

restricted to

is essentially the representation

rI

of Section 2.5

S-L~(~).

Corollary ~.20 associates to the trivial representation of GLI(~)

the representation

~--i/2 of

G-L2(~) and to the representa-

9S

tion

sgn(x)

of

GLI(~ )

the representation

this is the sought-after correspondence Concluding remark. Moreover,

rl(~ )

If

~3/2"

Modulo

(0,~),

W I.

F = @, rl(T)

is independent of

defines an ordinary representation

of

T.

SL2(@ ).

In

fact Kubota has shown in [23] that the "even" piece of

rl(~)

the complementary

s = 1/2.

series representation

of

SL2(~ )

Thus this complementary

series representation

the analogue of

The "odd" piece of

~I/2"

equivalent to the representation character

z ~ z/Iz I

SL2(@ )

once again appears as

rl(T )

for

F = @

is

induced from the

of the subgroup

operator is constructed fact that pieces of

of

at

is

rI

[[z W_l]~" The intertwining 0 z just as in Proof (2) of Theorem 4.19. The (over arbitrary fields)

could be identified

in this way was pointed out to Sally and myself by Howe; cf. his Zentralblatt

review of [i0].

w

Local Theory:

the p-adic places.

Throughout this Section, field of characteristic acteristic

of

F

F

zero,

will denote a non-archimedean

For simplicity,

will be assumed to be odd.

see the remarks after Lemma 5.6.

the residual charFor dyadic fields,

The metaplectic

cover of

G = GL2(F) (constructed

in Section 2.2) will be denoted by

~.

The purpose of this Section is to describe the local map S:T + for certain irreducible unitary representations non-supercuspidal 5.1.

~,

namely the

representations.

Basic Representation Theory. Suppose

of

of

~

~

is a (not-necessarily preunitary)

on a complex vector space

admissible if (i) for every is an open subgroup of group

~'

of

~,

~,

V.

Then

~

representation

is said to be

v c V, the stabilizer of

v

in

and (ii) for every open compact sub-

the space of vectors stabilized by

~'

is

finite dimensional. If

~

is irreducible and admissible,

if for every vector

v

in

(5.z)

we say

Y

is supercuspidal

V,

~ Y(u)vdu

= o

U for some open compact subgroup meaningful

V

of

N,

Note that

(5.1) is

since

~

splits ove#

Now suppose

Y

is an irreducible admissible non-supercuspidal

representation of in

U

satisfying

~.

Let

N,

V(~,N)

(5.1) for some

denote the set of vectors U

as above~

union of all its open compact subgroups a subspace of Since obtain in

B

Since

U, V(~,N~

N

v

is the

is actually

V. normalizes

V/V(~,N)

V, ~(~) preserves

a representation

~'

of

V(7, N). B

Thus we

(actually

~/N,

97

since ~'

~(n)v-v

e V(~,N)

will be n o n - t r i v i a l The significance

is that

V/V(Y,N)

cam be imbedded of

for all on

Z2

v e V

and

n e N).

if and only if

of our a s s u m p t i o n

(hence ~')

that

~ ~

is genuine

induced

on

~.

not be s u p e r c u s p i d a l

is then non-zero.

in the G-module

Note that

Consequently,

from an irreducible

quotient

V/V(~,N). To be more precise,

is irreducible,

v

the vectors

is an open compact is a finite

fix

a non-zero

7(~)v,

subgroup

set of r e p r e s e n t a t i v e s

V/V(~,N)

is a ~ - m o d u l e

subspace

V'

of

~

in

B\G/~',

w h i c h contains ~

let 6

of

define,

If

v,

Since ~'

and

{gi ]

under

also span

V.

That is,

Thus there is an invariant

in

and is such that the

V/V'

denote the modular

Ind(B,~,~),

V.

transforms

V(~,N)

B/N

V.

of

v i = ~(~i)v

of finite type.

representation

As always, imbed

V

span

in the stabilizer

of the finite n u m b e r of vectors

natural

g e ~,

vector in

for each

= W

is irreducible.

function v

in

for

B.

To

V,

fv(~) = T(~)v where

v

denotes

fv:~ ~ W

the class of

v

in

W = V/V'

Clearly

satisfies 1/2

(5.2)

where

T = 5 -1/2 ~.

between

~

That is,

and ~

subgroup If

of 7

= 6

Thus

v + f

Ind(~,~,T)

v

Ind(~,~,T)

(5.2)

is an i n t e r t w i n i n g

(non-zero

since

of

~

in addition

Ind(~,~,~).

to being right

operator

is not supercuspidal).

as the space of functions

(We are f:~ ~

invariant by some open

~.) is trivial

on

Z2

a general t h e o r e m of Jacquet irreducible

(~)T(~)fv(~)

is a. s u b r e p r e s e n t a t i o n

interpreting satisfying

fv(~)

admissible

this conclusion and H a r i s h - C h a n d r a

non-supercuspidal

is a special case of classifying

representations

the

of a p-a.dic

98

reductive

group.

Now let

T

denote an arbitrary

representation irreducible

of

B/N.

unitary

irreducible

(finite dimensional)

Henceforth we shall deal exclusively with

representations

of

~

which are equivalent

to

Ind(~,~,T) o r some u n i t a r i z a b l e representations, Let

B0

subrepresentation

have even v-adic order

subgroup

of

To a n a l y z e

such

the theorem below is crucial.

denote the subgroup

modulo units).

thereof,

(i.e.,

B0

In particular,

and its irreducible

B

whose diagonal

the diagonal

By Lemma 2.11,

B.

of

entries

entries are squares

lifts homomorphically

]0 = B0/N = A0 x Z 2

(finite-dimensional)

as a

is abelian

representations

are of the

form x

aI

(5,3)

TO(~I,~2)([O

with

~i,~2

quasichara.eters

Theorem 5.1. representation

of

is trivial on

a1

0 a2]'~)

~([0

other hand, and

[

a,2],C) = C#l(al)P2(a2) of

Suppose

~

~ = B/N

on a complex vector space

Z2,

V

T

is an irreducible

is one-dimensional,

= ~l(a')~2 (b)

if

F x.

for some choice of

is non-trivial

is equivalent

on

Z2,

V

finite dimensional V.

If

and

~i,~ 2,

On the

is four dimensional

to

r(ZI,U2 ) : Ind(go, g,~4D(~l,Z2 ) with

rO(~l,~2 )

a,s

in

(5.3)).

The first part of this Theorem is obvious. we shall exploit the theory of representations mal subgroup Clifford

of finite index.

for groups with nor-

This theory goes back to Weyl and

and has already been described

in Lemma 4.3.

To prove the second

for unitary representations

99

As before, If

L0

let

H

suppose

denote

the subgroup

H = G

Lemma

if

5.2.

representation

L0

of

L*

H(Lo)

(i)

Ind(H,G,L*)

(2)

the restriction

(3)

of

subgroup

irreducible h

in

G

representation

restriction

of

G. N,

(Lo)h = L O.

and equals

is any irreducible

whose

of index two in

such that

is self-conjugate

Suppose

is selfconjugate

N

otherwise.

finite-dimensional

to

N contains

L O. Then

is irreducible, of

Ind(H,G,L*)

and equivalent

is irreducible

L 0 | L h0

to

al__~ifinite-dimensional

if

L0

otherwise;

irreducible

representations

of

are so obtained. This

of

[3].

assumes

lemma is essentially Thus we shall not

that

G contains

this assumption

matrices units

whose

the content

include

the normal

diagonal

Thus we can prove

B+

of

units.

B

N

consisting

Clearly

5.1 by applying

properties

aI = ([0

and

wL(x)

but

o

a'

],~)(

[~0:~+]

[o I

symbol

o a~

it follows

)-I

],~'

0 a2]'~)(L2'(ai'a2)(a2'al))

al, a2, a~,a ~ e F x.

In particular,

fixing

in

= (m,x)~i(x).

L 0 = TO(~I,~2),

then

In other words, H(Lo)

= T 0.

= [~+:A] = 2.

Lemma 5.2 to these pairs.

of Hilbert's

a2

of

modulo

~(~i,~2 )h = ro(~i,~ ~) with

III

(In [3] Boerner

are either both squares

modulo

([ai o a1 o a~],C')([o

= ([o

12 of Chapter

of order two outside

subgroup

elements

Theorem

From elementary

for arbitrary

a proof here.

an element

or both non-squares

(5.~)

of Section

is not necessary.)

Now consider

h

is a normal

is any finite-dimensional

Clearly

G

N

if

N = ~0'

Consequently

G =

that

100

is irreducible of

~+

and all i r r e d u c i b l e

non-trivial

on

Z2

from

and the r e s t r i c t i o n ~i,~ ' 2'

(with of

of

(T:) h

Fixing h = ([0

is not equivalent

Z'

to

~0

is

to

Indeed

~nd(~+,~,~(~l,~

2)

representation

and Theorem 5.1 is proved.

to an appropriate, basis, to

~0

the

takes the form

0 0

0

0

o

(~,ala2)~

o

0

0

0

0

o

-rO(~l,~2)([O

T'.

],i) in

TO(~l,~2)|

Hence every irreducible

Ind(~0,B, T0(~l,~2))

a1

~'.

al 0

With respect

Co.r011ary 5.3.

(5.5)

representations

a2],C) = (w,a2)rO(~l, P2)([ 0 a2],~),

as above).

of

and

aI 0

~ is of the form

restriction

A+,A,

(5.4) that

zO(~l'~2)h([o

dimensional

are so obtained.

Now apply Lemms 5.2 to it follows

finite

a2],~)

8

~

(w, a,2) L~

0

o

(|

We close this S u b s e c t i o n with some Lemmas w h i c h will be useful in our analysis definition

of the representations

of the local map for

Lemma 5.4. representation of

A

Suppose of

~.

Clearly

~

in

is a f i n i t e - d i m e n s i o n a l

A

and in our

F.

Then its c h a r a c t e r

whose p r o j e c t i o n s Proof.

~

Ind([,~,T)

X

T

genuine

vanishes

on elements

are not squares.

is a class function

satisfying

T L for all of

~

~ c A in

A

and

C ~ Z 2.

is not a square,

L On the other hand, by Corollary 2.12

if the p r o j e c t i o n

101 for some

~ c ~.

Thus our claim is immediate.

Lemma 5.5.

The character of

T(~I,~2)

is computable

from

2 Corollary 5.3.

Indeed suppose

X

(5.6)

(a)

:

[ = [(~ 0

~0 2]'~)"

Then

11-~1(c~2)~2(,82 )

T(P-1, ~ 2 ) Proof.

HllbertTs symbol is trivial on squares.

Lemma 5- 6.

Suppo s e (i = 1,2)

~i(a) : ~i(~) for all squares Then

T(~I,~2 )

[]

a c FX~ equivalently, is equivalent to

(~i)2 = (vi)2

Fx .

on

T(Vl,V2) ~ in particular

multiplication by characters of order 2 is irrelevant. Proof.

Compare characters~

Concluding Remarks

[]

(the case of a dyadic field).

Suppose the residual characteristic of

F aI

is even, say

2 n.

Let

0

A0

denote the subgroup of matrices

[0

a2

have even

has unit part congruent to a

square modulo

v-adic order and 40.

Then

T

aI

splits over

irreducible representations

T

with the non-trivial character of in

~.

Thus

(cf. Theorem 5.1 where still valid.

Z

where both

(~l,m2)

?2).

But

has dimension

[FX:(FX) 2] = &).

aI

and

A O, and each of its

(non-trivial on

from a one-dimensional representation

[FX: (FX) 2]

a2 ]

Z2) of

]0

is induced ~0

(tensored

has index

~(2n) ~

Moreover,

Lemma 5.6 is

Since a detailed discussion of the representation

theory of the metaplectic group over a dyadic field will appear elsewhere, henceforth we shall deal exclusively with the case of odd residual characteristic remarks).

(except possibly for a few parenthetical

5.2. Class i representations. In this paragraph we begin a careful analysis of the induced representation Y(Ul,~2 ) = ZndG,~,T(~I,U2))o Denote its space by is non-trivial on Lemma 5.7. of

B(~I,~2 )

~(~l,U2).

The restriction of

Let

B(UI,~2 )

is admissible. space of

V(T)-valued functions on

aI x

tion

to the dense subspace

Then the dense subspace of K-finite vectors in

~(([o all

~(Ul,U2 )

V(L) denote the four-dimensional

consists of locally constant

for

~(UI,~2)

Z2.

~-finite vectors in Proof.

and keep in mind that

aI

aI x ( [ 0 a.2],C)

for

I/2T ( a I

a2]'~)~) = I ~22 I e ~.

~

B(UI,~2 ) satisfying

x

[o a2]'~)~(~)

On t h e o t h e r

hand, by Zwasawa's d e c o m p o s i -

G,

~ = ~ N K.

(5.6)

Thus this space is naturally isomorphic to the space of locally constant V(L)-valued functions on

K

satisfying

a.I x a.2]k) = T ( [ 0 a2]'l)M(k)

aI x

M([0

for all

a1

x

[0

a2 ] e B N K.

(Since

aI

and

a,2

are units in

F x,

aI

I~1 = l . ) To prove admissibility, let

H(~')

~' of

G

denote the subspace of K-finite vectors in

stabilized by

~'.

Since each

compact open subgroup

~' A ~,

values on the finite set finite-dimensional, proved.

fix an open group

~

in

H(~')

each such

K/~' n K.

~

and

~(~i,~2 )

is left fixed by the is determined by its

Consequently

H(~')

is

and the non-trivial part of admissibility is

103

Definition 5.8. class i

A genuine representation

of its restriction to

Y0: ( k , r

[

of

~

will be called

contains the representation

-~ C

at least once; equivalently,

its restriction to

K

contains the

identity representation.

and

2 Theorem 5.9. ~(~i,~2 ) is class i if and only if both 2 i D2 are unramified. In either case the identity representation

occurs at most once. Proof.

Each

~

in

B(~I,~2 )

aI x M([O a2]'l) :

(5.7)

for all

aI x [0 a 2] e B.

a I 1/2 I~I

Moreover,

(hence identically equal to

satisfies

~(e)

if

aI 0 r(~l, P2)([ 0 a.2],l)M(e)

e0 on

is right K-invariant K),

aI 0 T(~I,~2)([ 0 a2],l)~(e ) = ~(e)

(5.8) for all

aI 0 [0 a2] e A n K.

Thus

space of K-finite vectors in) satisfying

(5.8).

~(~i,~2 ) B(~I,~2 )

In particular,

is class i only if (the contains some function

by (5.8)

if and only if (5.8) obtains for some such

~(~I,U2)

is class i

~.

From the Corollary to the proof of Theorem 5.1, recall that aI 0 the eigenvalues of T(~I,U2)([O a2],l ) are

~i = ~l(al)~2(a2 ) h 2 = (~,ala2)~ I h 3 = (|

I

h 4 = (|

1.

and

Thus

~(~!,'~2)

is class I if and only if one of these

hi

is

1

104

for all F x.

al, a 2 c 0 x = U.

Hence

(5.8)

But

(~,.)

is impossible

On the other hand,

suppose

Then by Lemma 5.6 we may assume In this case,

~i ~ i

Note finally Indeed

these

the space

eigenvalues

one-dimensional space

consists

constant V(T)

Analogous

uI

and

in

of

those

a ramified 2 o__rr ~2 2 ~2

and ~2

are unramified.

is class h~

i.

are non-trivial.

(5.8)

of the theorem whose

of

F x.

comprises

Thus a

is complete.

restrictions

in the one-dimensional

to

subspace

(This

K

are of

~i. )

results

hold for dyadic

of

is ramified.

characters

satisfying

~

character

are also unramified.

P(Ul,~2 )

ramified

~(~i,~2 )

and the proof

2 ~i

~2, h3, and

then define

precisely

to

2 ~i

both

~i = i,

and take their values

belonging

GL 2 (0).

space

if either

and consequently

that when

of functions

defines

fields with

KN

replacing

5.3. Hecke operators. Let

H(~,~ 0) denote the Hecke algebra of continuous

supported functions

~

(5.9)

on

r

for all

[,~T

in

G

compactly

satisfying

g ~') : Yo(g)r

K.

Multiplication

in

H(G,~0)

is given by the

convolution product (5.i0)

r

Note that if r

<0

: {r

: ]' 41([ y-l)r

is any continuous

function on

if

(satisfying

r

spherical,

r

~) = Y0(~)m0(~)),

~O*r

In

B(~I,~2 )

is also zonal

and = X(r

i e H(Z,~O).

all

Our p u r p o s e

in this 4.

Subseetion

important

special

contained

in the two lemmas below.

Lemma 5.10.

is

(5.Zl)

Fix

[([0

to

n

and

m

in

~.

n

and

Proof.

w

m

w

w m]'l)[ =

in

U

C~z2

n

K([ 0

some

0 wm],l)K(l,{)

n,. s,n,d,,m

,a,re,,ev,en

0nly the second assertion is non-trivial.

Fx

for

are squares).

it, suppose first that r

h(r

Then

and th,e' union is disjoint if, ,and only, if w

eompute

The required double coset space analysis is

Cn 0

unit

satisfying

is any zonal spherical function in

~O*r

(i.e.

~

the integral in (5.10) is still well-defined.

particular,

for

[.

n

= w.w2%

such that

with

g c g.

(w,c) = -i, then wn 0 ]

z o l)([ ( [ o c]' o

|

'

l)(l,-l)

To prove

If we choose a

108

m]'l)([Oi 0e ] ,I)

: ([ n 0 This implies,

however,

that

wn x([

~

Cn

0 ]

0

m

: x([

w

0

.

Thus the union in (5.11)

,1)x

is not disjoint if

Suppose now that both

wn

and

wm

.

n (say) is not even.

are squares.

If the two

sets on the right side of (5.11) are not disjoint then they coincide. Consequently (5.12)

(kl, l)([ wn 0

0m],l) w

for some

k I, k 2

K.

Thus

~([ w n 0

in

= ([ wn 0

~m],l)(k2,!)(l,-l)

But now both

n

and

wm

are squares.

0 ]),k2) = ~(kl,[W n

Om] ) = l, and (5.12) implies

wm

w

0 n

(kl[ 0

n

wmO ],i) = ([Wo

an obvious contradiction.

wOm]k2'-l)'

[]

According to this lemma, a non-trivial element of n cannot be supported in ~([w 0 ] I)K unless both n 0 wm ' even. In particular we have: Corollary 5.11.

if

~

~

K([

Cn,m(~ ]._. = 0

both

n

and

m

and

m

H(~,~0).

wn

o

0

m

]

,1)~(1,~)

otherwise However,

it is identically

zero unless

are even.

Corollary 5.11 explains why Shimura and his predecessors treating forms of half-integral

weight)

operators

to square

Tk(%)

corresponding

and p.450 of [38]; Shimura's F = ~.)

are

The function

{~ always belongs to

H(~,~ 0)

T(p 2)

(in

dealt mostly with Hecke %.

(Cf. Proposition 1.0

is essentially

r

Lemma 5.10 appears as Proposition 7 in Kubota

with [22].

For

107

even

v, and

K = K N, 41, 0

is not necessarily

zero;

again,

compare

p.~5o of [38]. The collection algebra

H(~,~0)

of elementary

[~n,m~,

of

with

n

a%-spherical

divisors

for

and

m

even,

functions.

generates

the

Indeed by the theory

G, n

(5.13)

~=

u~([~

On the other hand, Thus by

(5.9),

n,m

0

each

r

m]' l)~ "

in

H(~,~O)

is compactly

supported.

@

is completely determined by its value at a wn 0 ] finite number of matrices ([ ,i). That is, each such $ 0 wm a finite linear combination of the functions Sn,m of Corollary Using a modification

show (Cf.

@n,m*@n,,m,

of the usual

= Cn,,m,*@n,m,

Section 4 of [21].)

eigenvalues

Our interest,

for the zonal

Lemma 5.12.

hence

spherical

The double

arguments, that

functions

in

5.11.

one can directly

H(~,~0)

however,

is

is abelian.

is in computing ~(~i,~2).

coset 2

% = is the disjoint for these

union of

cosets

~2

[o

~ q2 + q

left K-cosets.

(with

z ~ (0/~2o) x)

Representatives

are

o] ,

[1 02][_ 1z lO] 0 and

2 [0 with

y = O,c,

i 01[o m~ [| c 2,...,eq-I

(q-l) st roots of unity This include

in

lemma concerns

a proof.

1 0 ]' c O/W0

and

c

a generator

of the

F x. only

G

We do, however,

and is classical. include

Thus we do not

the following

remarks.

108

In

-G,

([c w

-i 2-k

for each unit

0],l)([wkc -w

c

c

2

in

F.

0

0

0

-i 2-k ]'(c'w)k) w c

(This follows from elementary computations

involving Hilbert's symbol.) (K[1

: ( [w k

1],l)

Thus if

z = wkc,

1 ,1)

[ z

o w2 ] -i

o]

= K([ w~ 0

c-i

W2 - k ] '

(C' w ) k ) "

On the other hand, w2 (K[ 0 0

1 , i) z][~y1 01][~ o]

: (K[Zo 20][0Z-~Y]'I) Theorem 5.13 . transforms under

Suppose

~

~(~i,~2)

according to

~0 "

is class one and Let

~(w 2)

"Hecke operator" corresponding to convolution by

~(|

~0 {Z(~I'~2)

denote the

@0,2'

and define

by

9 (|

= x(~)~o(~) S.

Then if

~i(x) = Ixl i l(w) = q(q 2s I + q 2s2)

Proof. ~i

By Lemma 5.6, there is no loss of generality in assuming

unramified.

Also, by our hypothesis on

~0~

~O((k, 1)) ~ V o where

T0

is some distinguished vector in

V(T).

T0

belongs to the subspace of fixed vectors for

to

A N K.)

Thus

(More precisely, T(~I,~2)

restricted

109

l(w)V 0 : ~0*~0,2(e) =

f

~0(y-1)90,2(Y)dY

Z2\~ = ~ %(Y-1)dy H Now we apply Lemma 5.12.

According to the remarks following

this Lemma,

w2

~(~)v o : ~ o ( ( k [ o +

z

z:~kc

~]k)-l)dk -k

[%(([| ~

-i

-c , wk_2] ( (| ~) k) (k, 1) ) dk

o

+ Z ~ ~0(([ 10 wy_2],i)(k, 1)dk But

~0

belongs to

T(~I,~2 ).

Thus we have w-2

~(W)Vo = I|

)([o

i~ w-k

+

(c|

-k,liz([

0~zE=~kc

0

-i -c I)VO ~ k-2] '

(modw20)

+ I|

~

L([ I

yeO/mO

0

~[2],l)Vo w

To continue, we need to further analyze the second sum appearing above. First consider the set of non-trivial coset representatives of

0

modulo

q(q-l) 9 kc

units.

with

~20.

This set has

The remaining

q2-1

q-i

elements and contains

representatives are of the form

k = i.

Now consider the sum

z

(c,~)Z([ ~

z=~e (modw20) Since

(q-l)

is even, and

-I

0

(r

{z]

-i

-e-i ]' l)Vo

= (-i) g,

this sum actually

110 -i _c-i (Z([ w w- I ],i)~ 0 does not depend on c.) Thus the 0 second sum appearing in (*) simply equals q(q-l) lwll~2(w-2)~ 0 and

vanishes.

x ( ~ ) v o = q~l(~-2)Vo . ( q - l ) ~ 2 ( |

, ~2(|

o

.

That is,

~(| = q(q 2s I + q 2s2) , as desired.

[]

In Subsection 5.4 a local map Sv:~ -~ will be defined for irreducible quotients of

~(WI, W2 )

and Theorem

5.13 will be used to explain how this map is consistent with Shimura's correspondence. for

In the meantime,

recall the analogue of Theorem 5.13

G = GL2(F ).

If

p(~l,~2 )

denotes the representation of

G

induced from

the character aI

x a2] ~ ~l(al)~2(a2)

[0 of

B, p(~I, Z2 )

is class

1 iff both

Z1

and

~2

are unramified.

s.

Moreover,

if

~i(x ) = Ix I i, and

K-fixed function in

l[O

(5. m2)

B(ZI,~2),

~0

is the (essentially unique)

then

%(hy-l)dy = ql/2(q si + q s2 )%(y)

K[ o ~]x Note that the operator on the left side of (5.12) is (essentially) Hecke's classical

T(p).

For details,

see [7].

In Section 6 we shall also see that Theorem 5.13 is consistent with Kubota [21], P.53.

(q2 + q)~(l,|

Indeed our

T(w 2)

is Kubota's

5.4.

The Local Map In this Subsection we shall define the local map

for irreducible representations

of

~

imbeddable in

We begin by presenting a useful

(if not unexpected)

S: ~ v ~(~i,~2 )

character

computation. Let

H(G)

denote the algebra of all locally constant compactly

supported functions on in

H(~)

~

satisfying

is given by (5.10).

representation

of

~

The product

is any admissible genuine

~ ,

=

f ~ ~(f

defines an "admissible" In particular,

If

f(~g) =of(g).

~(f)

Theorem 5.14.

~ f(g)~(g)dg z2\~

representation

of

H(~)

on the space of

~.

has finite rank. If

f e H(~),

trace(~(~l,~2)(f))

=

then ~

f(g)~(~l,~2)(g)dg

z~ where

~(~i,~2 )

conjugate

in

~

denotes the character of

~(~i,~2):

If

g

is

to some aI 0

([ 0 a.2]'~) with both

al, a 2

squares in

'

F x,

then

ala2 21m/2 (al-a 2) otherwise,

~(~) ~ 0 .

Proof.

Abusing notation,

we shall confuse

~(~1,~2 )

with

the subspace of functions which, are right invariant for some open subgroup of

~ .

finite rank on ~(WI,~2)

Then, as already remarked, B(~l,~2).

and the fact that

~(Wl,P2 )

is of

(This follows from the admissibility f

has compact support.)

Hence the

of

112 trace of

~(~I,W2)

Now if

certainly exists.

~ c~(pl,~2 ),

z2\~ =

~

~(~)f(~)d2

=

S

~ (2)f(k-l~)d~

K Z2\~

=~ ( [

8~([)l/2~([)f(k-l~kl)d%[)~(kl)dkl

,

K z2\~ where

a1 0 a1 6T([ 0 a2 ] ' ~ ) = [ ~ l ,

Haar measure

on

~ ,

~=T(~l,~2),

and

dg~

is a left

Thus

(Y(f)~)(k)

: SKf(k,~l)~(kl)dk

l

K

where

Kf(k,k l) =

~

6~(~)l/2~(~)f(k-l~kl)d~

.

z2\~ One can check now that the integral operator space of all locally constant functions from

K

in

on

Z(~I, Z2 ).

Therefore,

precisely the trace of

the trace of

K,

~(f)

K, to

actin 6 on the V(~),

has range

B(~I,~2 )

is

i.e.

try(f) = ~ tr(K(k,k))dk K

= ~

6Z(~)I/2tr(T(~))( ~ ~ f(k-l~nk)dkdn) d~.

z~ ~ To continue,

K N

observe that -aln

al O

([0

a2]'~)([0

I n

1]'1) =([

aln

-

0

al a2]'l)([0 I

Thus, making the change of variable

aI 0

a2]'~)([

0

al-a2]'l) i

al-a 2 n ~ (-~------)n yields the formula

~

113

try(f) = ~ tr(T(~))l(al-a2)2[1/2(~ Z2\~

ala2

~f(k-ln-l~ nk)dkdn)d~ . K N

To show that this actually equals

we shall compute this last expression directly. Recall the well-known integration formula

ShIgldg : : I

:Ial

G

hIx laxldx AXG

Here the summation extends over all equivalence classes of Cartan subgroups

A

of

G,

and

a(a)

[ (al-a2)2 =

ala 2

I

Applying this formula (keeping in mind that the central function ~(~i,~2)

vanishes off

A2),

we have

z~: f(:):(~l'~2)(:)~ ZGf(g'll:(~l'~2)(g,l)dg =

= ~l ~2 A(a)~(~I'~2)((a'I))(A~G $ f(g-lag'l)dg)da "

However, since

~(n,k) = ~(k,n) = i, this last expression equals

i2 ~ A(a)~(~l'~2)(a,l)($ $ f(k-ln-l(a,l)nk)dndk)da A2

N K

= 2 ~21 (al-a2)21ala2i/2[~l(al)~2(a2) + ~2(al)~l(a2 )]($ Z f(k-!n-l(a'l)nk)dndk)da NK

=4 A $21(al-a2)21!/2~l(a!)~2(a2)($ ala2

$ f (k-ln-1 ( a, 1) nk) dndk) da

N K

114

T complete the proof, recall from Lemma 5.5 that aI 0 ~l(al)~2(a2) = ~ tr(L([ 0 a2],!)) when al, a 2 are squares in F x. Thus

z2\~

f ( Z ) Y ( % , ~ 2) ( 2 ) ~

=

tr(~(~) )I (al-a2)2 ala2 11/2(~ ~ f(k-ln-l~ nk)dndk)d~ z2\~

~ = tr T ( f ) ,

as claimed.

[]

Theorems 5.13 and 5.14 suggest that we define a map follows.

If

Y

is equivalent to

~(~i,~2),

set

~ ~ v

~ = S(~)

as

equal

2 2 p(~l,~2). The problem with this simple definition is that 2 2 p(~l,~2) and/or T(~I,~2) may be reducible. to

In [18], Jacquet-Langlands

establish the following results:

(a)

If

~l~21(x) ~ Ix I

or

Ix[ -I, p(~l,~2)

(b)

If

~l~21(x) = [xl -I, p(~l,~2)

is irreducible;

contains just one sub-

representation and it is one-dimensional; (c)

If

~l~21(x) = Ixl, p(~l,~2)

representation, If

this time infinite dimensional and "special".

p(~l,~2 )

not, ~(~i,~2 )

again contains one sub-

i__ssirreducible,

it will be denoted

~(~i,~2);

denotes the irreducible subrepresentation

if

of p(~l,~2).

In either case, the obvious equivalences obtain. (d)

If

~l~21(x) J Ix[

or

[xl -I, ~(~i,~2)

is unitarizable

~2

are unitary (in which case

if: (i)

both

~i

and

is a continuous series representation)

(ii) (in which case

~l~z(x) ~(~i,~2 )

~(~i,~2 )

o_~r:

= Ixl ~ ~2(x) = ~Z-lT~-l, and

0 < o < 1

is called a complementary series representa-

tion). If

~l~21(x) = Ix[

or

Ixl -I, ~(~i,~2)

is unitarizable iff

115

its central character is already unitary; is called a special representati0n

in this case, ~(~i,~2)

(if it is not one-dimensional).

Using [i0] one should be able to obtain analogous results for ~: (a)'

The representation

~l~21(x) ~ Ixl I/2

(b)'

If

representation,

or

~(~i,~2 )

is irreducible if

Ixl-i/2; in this case, ~(~!,~2 )

~l~21(x) ~ Ixl -I/2, ~(~i,~2) which we still denote by

is denoted

contains one sub-

~(~i,~2);

this representa-

tion is always infinite-dimensional and sometimes class i (namely when ~ and ~22 are unramified and v is odd); (c)'

If

~l~21(x) ~ Ixl I/2, ~(~i,~2)

one subrepresentation,

a special representation

representation is infinite-dimensional (d)'

The representations (i)

~i

again contains just

and

~2

~(~i,~2);

this

but never class I;

in (a)' are unitarizable if either:

are already unitary (this gives the

continuous series); or (ii) ~2(x) = ~i-~-~ -I, and

~l~21(x) =

x ~, with

0 ~ a ~ 1/2 (this gives the complementary series).

The representa-

tions in (b)' and (c)' are unitarizable if and only if unitary.

Moreover,

~i~2

is

the obvious equivalences obtain.

Now we make the following definition.

Definition 5.15.

The map

is defined for irreducible unitary representations by setting

S(~)

equal to

Proposition 5.16.

~( 2

The map

class one representations

~ = ~(~i,~2)

2). ~ ~ ~

is one-to-one and takes

to class one representations.

116

2 2 2 2 Proof. If Tr(~l,~2) = Tr(~I, v2), then by (e) above ( 2, 2) 2 2 2 2 = (~i,~2) or (v2,Vl). In either case, T(~I,~2) = T(Vl,~2)

by Theorem 5.14. Thus S is one-to-one. On the other hand, 2 2 2 2 ~(~i,~2 ) is class I iff ~I and ~2 are unramified, so ~(~i,~2 )

is class i iff

Proposition 5.17 .

S(~)

is (Theorem 5.13).

The map

S

[]

is natural from several points

of view, namely: (a)

S

takes special representations to special representations

and "trivial" representations (b)

S

(c)

Roughly speaking,

preserves eigenvalues for the Hecke operators;

Proof.

S

preserves characters.

A "trivial" representation for

is a one-dimensional IxI-i/2).

to "trivial" representations,

representation

GL2(F)

(resp. a

~(BI,~2 )

so

~0,2

F v=~p.

BI~2 I=

Let

k

T(p 2)

that the global field is

denote convolution by I 0 (resp. the characteristic function of Kp[ 0 p]Kp) Now, as

always, fix

p k / 2 - 2 T(p2)

Shlmura [38], p.450. function for

(resp. T(p))

to be an odd integer.

denote the operator

(i.e.

with

Thus Part (a) follows from (b), (c), (b)' and (c)' above.

To prove (b), suppose (for simplicity) Q

(resp. G-L2(F))

Finally,

B( ~i,~2 2 2)

~2(x) = Ixl2Sl).

Tk,~k_i/o(p 2)

(resp.

let

when both

Let

Tk/2(p2)

S(~00) denote "the" K-invariant 2

and

~22

regular, put

and

p(~l,~2) 4+(u

are unramified

Then by Theorem 5.13, and the remarks

To explain (c), let representation

Tk_l(p))

p 2 -i T(p)); cf.

immediately following it, the elgenvalues for respect to

(resp.

Tk_l(p)

G.

and

S(~ 0)

with

are identical.

X(~I, B2) of

~0

denote the character of the If

yeA

is such that

equal to !(yl+Y2)2/yiY2)[ I/2.

Theorem 5.14 and its well-known analogue for

G ,

72

Then by

is

117

(Recall

g(y)

denotes the lift

Remark 5.18. Sk/2(N,x)

to

respect as well.

.)

[]

A crucial feature of Shimura's map from

Sk_I(No,X 2)

the Hecke ring.

y ~ (u

is that it preserves

eigenvalues

Our map is consistent with Shimura's in this

for

5.5

The Basic Well Representation. As always,

F

is a non-archimedean

residual characteristic

and

additive character of

F

T

local field with odd

is the canonical non-trivial

whose conductor is

OF .

Fix

q =ql

"

As in Subsection 4.4, we shall not deal directly with the Well representation

rql.

the representation

Rather we shall deal with

of

~-L2(F)

in

L2(F)

rl(T).

This is

given by the formulas

(5.13

rl(T)([ 01 ~])~(x) = ~(bx2)~(x)

(5.14

rl(T)(~)%(x ) = y(ql, T)}(x)

and (5-15

r!(T ([a0 a-01])@(x)

Note that

rl(T)~

is even (resp. odd) if

Theorem 5.19.

Let

re

the space of even functions odd functions.

: (a,a)lal I/2 ?(ql'Ta)u ) %(ax)

%

is even (resp. odd).

denote the restriction of in

L2(F)

and

.

r0

rl(T)

to

the restriction to

Then:

(a)

rl(T) = r e a r 0 ;

(b)

re

(c)

(R. Howe)

(d)

re

and

r0

are irreducible; r0

is supercuspidal;

is equivalent to an irreducible quotient of the class

one non-unitary principal

series representation

p(i/2,1)

of

~Z2(F) Proof (Sketch). (a)

Obvious.

(b)

Formulas

commuting with

(5.13)-(5.15)

rl(T )

imply that the only operators

are the identity operator and the reflection

operator (x)

+ ~

(-~).

119

But this latter functions (c)

in

operator

L2(F).

Hence

To prove

for all vectors

acts

r0

v

N

trivially

II

in the space

of even or odd

(b) is immediate.

is supercuspidal

in the space of

it suffices

to prove

that

r O,

rO(u) v du = O, N

i.e.,

for all odd

~

in

L2(F)

and

x e F,

(5.16) F If

x = 0

this is obvious.

So suppose

x % 0.

Indeed

Recall

~(x) = 0

from Example

by the oddness 2.29 that

equivalent to rl(~t2) for any t e F x. This implies 2 x lies outside O F , i.e., ~(nx 2) is non-trivial on

s(nx2)dn

of

rl(~)

~.

is

we may assume OF .

Thus

= 0

F and the proof

of

(d)

A

Let

(c) is complete. denote

the inverse

a

triangular p(I/2,1)

subgroup

image

in

S--L2(F) of the upper

x

[[0

is the genuine

a -1]]

of

SL2(F).

representation

By definition,

of

~-L2(F)

induced

from

the character [[a Xl],~] ~ ~ [ a I I / 2 [ ( T , _ X ) ( T , T ) I / 2 ] [ I - ( ~ , X ) ] / 2 0a of

~I "

For more

Since and

details,

the residual

p(I/2,1)

(d) it suffices

zonal

spherical

computation. function

of

those made

characteristic

are class

prove

OF .

to compare For

the K-fixed

For

in the proof

of

i with respect

functions.

Indeed

see [I0].

p(I/2,1),

F

to

is odd, K=SL2(0F).

eigenvalues re

this

vector

re

Thus to

for the corresponding

is a straightforward

p-adic

is simply the characteristic

the computations

of Theorem 5.13.

both

[]

are similar

to

120

Alternate proof for (c). of

An irreducible admissible representation

SL---2(F) in some vector space

V

trivial N-morphism exists between For

rI

in [0 1 ]

is super-euspidal iff no nonV

acting in the Schwartz-Bruhat

and the trivial N-space space

~(F)

the action of

is given by multiplication by the quadratic character

i.e., the only N-invariant functional on supported at the origin. archimedean

F)

is

V

x(nq(x)),

is a distribution on

But the only such distribution

9(x) +9(0).

C 9

(for no___nn-

Since this is trivial for odd

the proof is complete. Alternate proof for (d). that

9(x) ~r(g)}(0)

Following (c), use (5.15) to check

intertwines

r~

with the desired principal

series representation. Concludin@ remarks. for dyadic fields too.

F

The obvious analogues of (a) -(d) hold

4,

w

Global theory and odds and ends Throughout this Section

field. on

By

L2(X)

X=Z~GF~G ~

for all

g e GA

F

will denote an arbitrary number

we denote the space of square-integrable which are genuine,

and

~ e Z2 .

i.e. such that

~(g~)=~(g)

The regular representation of

in

L2(X)

is denoted by

T .

Also

L~([)

of

L2(X)

consisting of cuspidal functions and

L~(X)

~

denotes the subspace L~(~)

the subspace generated by poles of Eisenstein series on Let

functions

denotes ~

.

denote the direct integral with respect to Lebesgue

measure of the spaces of genuine principal series representations R(~,s) L2(y) c

with

Re(s) = 0

and

Im(s) >0.

is isomorphic to a subspace of

decomposes discretely.

According to Section 3, L2([)

whose orthocomplement

In fact

= T o 9 ~E 9 T c where (i)

T0

denotes the restriction of

T

to

L~(~) ;

(ii)

TE

denotes the restriction of

T

to

L~(~) ; and

(lii)

Tc,

the restriction of

continuous spectrum of

to

L~(X) , exhausts the

~ .

Since the decomposition of T 0 9 TE,

T

~

the discrete spectrum of

is known, T .

it remains to describe

6.1.

The discrete

non-cuspidal

For convenience,

spectrum

replace

X

by

~* = z~ ar\~ ~ The corresponding ~

L2(X*I

in

decomposition

--*T E

representation

denotes

(9 --*T E (9

T*c

the discrete

Fix

non-cuspidal

spectrum

T =~ T v v each place v of

be any non-trivial

character

F

denote

representation

S-~2(Fv)

of

the restriction L2(Fv)

.

-I

Z2~

v

Tv

(2.45)

produces

character

of

of

L2 (Fv)

in

on

0v

F~,/A .

r;(ql,T V)

for almost

For

r;(ql,Tv)

Then inducing

representation

in Subsections

representation v .

above,

Well

denote

every

in

v,

v 9

Thus we can a s s u m e

i for all odd

7"

the corresponding

Let

every

that

Fx .

fixed

of

of

to the space of even functions

i for almost

v

v

The induced is class

in

is trivial

a ~enuine T

F

rv(qI,Tv)

it follows

is a square

independent

let

rv(ql,~v)

is class

From if

of

Since

r;(ql,Tv)

of

is

~*_ = ~

Here

of the regular

is trivial

on

r;(ql,T v)

to

of

~v

m

is the canonical

v

which

[O101 ]

is

4.4 and 5.5. denoted

Thus the tensor

simply

e rv(ql),

product

e

re = | makes of

sense.

%

Theorem

morphic

r

In particular,

trivial

equivalent

rv(q I)

on

6.1

to

re

Subsection

(cf.

[9],

[I0]).

a genuine

The representation

In particular,

of the basic 2.6.

defines

representation

Z~ .

form of half-integral

a translate

e

weight

every square which

theta function

T~

integrable

is auto-

is not a cusp form is

e(z)

in the sense of

123 Proof local.

(Sketch).

Indeed,

The assertion

suppose

~ = | ~v

the theory of Eisenstein

series

with

=

results of Subsections r~(ql)

So since

sketched

r~(ql)

of

only

By v

representation But the local

"

imply that such

%

occur in

is easily shown to be irreducible,

TE

the local maps Suppose

representation

series

2 2 ~i

and

(not just S

~E )

introduced

v

~ = | %

is explained

in Subsections

is a constituent

is again a quotient of some non-unitary

v

T~ .

the

[]

The structure

as follows.

11/2

of

in Section 3, each

principal

4.4 and 5.5

Theorem follows.

together

1

is essentially

is a constituent

is a quotient of some non-unitary

yv(~!,~2 )

to be proved

~v(~l,~2 )

restriction

with

Wl~21(x)

2 2 ~1~2

imposed on

is

of

Then

series But now the

~ 21 and

that

4.2 and 5.4

~E "

principal

= [xl I/2

by pasting

~22

b o t h be

unramified. Let of

pv(Vl,V 2)

GL2(F v)

that

with

denote the usual principal VlV21(x)

2 2) --~v(~l,~2

Sv(%)

= Ixl

The definition

of

Sv

is the class one quotient

of

pv(Vl,V2)

Thus the theory of Eisenstein S(~) = | S v ( % ) a constituent constituents precisely, of

S

~i

and

series for

is an automorphic of

of

TE TE

l.e.,

if and only if

GL 2

the map

S

vI

7(Vl,V 2) and

v2

of

TE

of

~,

namely

TE .

More

will be in the image

are globally

squares of some

~2 "

the eigenvalues

of certain

F

imaginary number field and

is a totally TE .

series

of

is a bijection between

The theory above also ties in with Kubota's

of

is such

implies that

representation

and certain constituents

a constituent

series representation

real analytic

For each archimedean

representation

of index

v, 1/2.

~

v

computation

Eisenstein

series.

of Suppose

~ = | ~v is a constituent v will be a complementary

(Only in this case is

124

pv(~l,~2 >

irreducible if

~lU21(x)

= Ixl I/2 .)

For

v

finite,

will almost always be the class 1 quotient of the principal v series representation pv(Ixl I/4,

~xj-i/4)

Now suppose at

s = 1/2

E(g,~,s)

generates

is the Eisenstein series whose pole

~ .

Then the eigenvalues of

with respect to Hecke operators in

H(G-L2(Fv),O O)

E(g,~,s)

can be computed

in terms of the spherical function theory of the local representation ,

i.e., from Theorem 5.13.

Kubota's results for

E(u,s)

The results are in agreement with (ef. Section ! and [21], P.53).

6.~

Ce.nstructlon of cusp forms of half-lntegral weight Fix

F~Q

and

~ =AQ.

Let

(~A)gen

denote the set of

equivalence classes of irreducible unitary genuine representations of

~

and

G~

the set of equivalence classes of irreducible

uultary represeutatlons of certain

~

in

(GA)gen

G& 9

From Section 3 it follows that

correspond to classical modular forms of A

weight

k/~ ,

It is also well-known that certain

correspond to classlc~l modular forms of weight

~

in

G~

k-l.

Motivated by the correspondence between forms of weight ~nd

k-I

k/2

discovered by Shlmura it is natural to introduce a general 4

correspondence between

(Gg)gen

and

G~

following the local theory

developed in S~ctlons h and 5.

Thus we let

of

(irreducible unitary genuine

l-1

maps between

represeutatlons of of

G~ )

(Gp)gen

~p )

Gp^

and

~Sp]

denote a family

(irreducible unitary representations

with the following three properties: A

(i)

for each

~

(~&)gen'

S(~) =@ Sp(~p) determines a well-

deflued irreducible unitary representation of (ii)

for almost every

~,

S(~)

G~ ;

is automorphic if

~

is

autemorphlc~ (iLl)

if

~p

is not supercuBpidal,

in which case

naturally ~s~oclated to some pair of quasl-characters of of

x Qp~ Gp

Sp(~p)

is ~Imply

~(W~,~),

associated to the pair

with

~p

(Wl,~2)

the irreducible representation

Sp

preserves class 1

~e~reseut~tlon~ (Proposition 5.16) and the fact that | ~p

is

2 (~1,~2).

Implicit lu (1) is the fact that

f~ctored as

~p

class I for almost every

~

can be p

(Theorem

B.~), The necessity of the word "almost" in (ii) results from our having defined "automorphic representation" rather rigidly. we not required automorphic forms to occur in have to worry about

S

L2

Had

we would not

sometimes taking square'integrable cusp

126

forms on

~&

to slowly increasin~ non-cuspidal forms on

G~

See Conjecture 6.2 below and the remarks directly following it. Note that if

~

and

S(~)

ar~both

cusp forms then the "strong

multiplicity one theorem" of Casselman [4] and Miyake [27] implies that the family

[Sp]

is actually uniquely determined by conditions

(i)-(lil). To prove that at least one such family

[Sp]

exists one has

only a finite number of "bad" primes to worry about. every component ~%)

S(~p)

%

of

~

is non-supercuspidal and (for such

is defined as in (iii).

form is used.

To prove (ii) the Selberg trace

We shall treat these matters elsewhere,

in the near future.

In the meanwhile,

why constituents of

--*T O should occur in

we use r3

Note first that the restriction of forms must agree with S

or

Sp

of

below to explain r1

G

Indeed by (iii)

~k/2

of

Zp

for

corresponding to

T(p 2)

S(~)

to the

Furthermore,

~p

and

S(~p)

are class

corresponds to a classical modular form of weight

elgenvalue

~

according

preserves eigenvalues for the zonal

spherical functions as soon as ~

Vk_ 1

hopefully

to holomorphic cusp

takes the discrete series representation

to Proposition 5.17,

for

S

S

Shimura's correspondence.

discrete series representation

if

Indeed almost

1 . k/2

So with

(cf. Shimura [38]) the "new form"

is of weight

k-i

and has eigenvalue

P

T(p). Now recall the correspondence D3:~ ~ ~ .

This correspondence is defined on a subset of in ~*.

(G~]gen

For convenience, ~

For each place

rp(q3,Tp)

sentation of

S--L2(%)

let in

and takes values

we restrict our attention again to

In fact we reformulate p

G~

L2(~)

in the context of

S--L 2

as follows.

denote the Wei! repre-

attached to

q3"

We assume in

127

advance

we have fixed a character

T (x) = e ~ix it follows is class

Since

Tp

is trivial

from the discussion

one with respect

According expounded

T = ~ ~ on

to

SL2(0p)

{\A

with

for almost

of Subsection

to the philosophy

in Section

of

P 0p

2.6 that

for almost

every

P

rp(q3,T p)

every

p.

of the Well representation

2 the representation

r3(T ) = | rp(q3,T p)

is a well-defined algebra

is generated

PGL2(~)

in

representation

by the adjoint

L2(~).

one obtains of

genuine

We denote

and irreducible

By inducing constituents

~

map we call

~.

action

between

Z~ \ G ~

of

A

action by

irreducible of

we obtain a map

and constituents

The non-trivial

whose

commuting

of the orthogonal

representations

to

S-L2(A)

this adjoint

a i-i correspondence

r3(~ )

of

~

GA

fact about

A.

which occur in

r 3.

D3

Thus

constituents

(no longer of

group

i-I) between

This is the

we use below

is

that

(6.1)

D3(s(~))

whenever

the left-hand

=

side makes

sense,

i.e.

S(~ ~)

occurs

in

A . The analogue section 4.3. p .

of (6.1)

One can also check

In general,

one wants

8(~,~)(g)

generate Here

:

a non-tzivial

~ e .<~(F~)

: S(~).

For

form of weight

"at infinity"

and v k-i

~

was established

directly

in Sub-

for all unramified

to show that the functions ~v(h]

Z

~c93

subspace ~

(6.1)

of

denotes

corresponding

r3(g)%(ho~)dh

L2(~)

which

realizes

some spherical

to a classical

such a result was obtained

functicn

holomorphic in [41].

A.

~

9

for cusp

128

Conjecture 6.2. 7~ = ~k/2 r3 .

and

Suppose

k~5,

Then

In other words,

k~5,

~ ~

is a constituent occurs

with

in the Well r e p r e s e n t a t i o n

every classical

arises as a theta-series

--* TO

of

cusp form of weight

attached

to a quadratic

k/2,

form in

3 variables. Sketch of "Proof". to conclude

that

trivial on

ZA

(If

S(~)

k=3,

The hypothesis

S(~)

occurs

in

and corresponds

k~5

TO .

should make

it possible

More precisely,

S(~)

to a cusp form of weight

k-l.

need not be cuspidal;

cf.

Shimura

is

[38], pp.458

and 478). Now I want to conclude I need a generalized that any component square-integrable

S(~)

also occurs

Ramanujan-Petersson ~p

of a cusp form

The truth of this assertion

consequence

of Deligne's

"everywhere" suggested

S(~p)

should

should occur in representation

implies by

proof of Weil's

to me by Langlands

is already

in

71/2 .

such forms,

with

representation

for almost

of

Gp.

(Cf.

in

Lipsman

A,

i.e,

occurs

in

Indeed

of weight

[50].)

[5],

D(S(~))

is a

That the

methods was

conjecture

is that of

in the usual For

p = ~

Gp/Zp

regular

this fact

it for all

is defined.

p

Therefore,

r3. 1/2

can not p o s s i b l y component

of

r3

occur does not

to ask how one can construct

or similar forms of weight 3/2, w h i c h

My conjecture

Gp

by Deligne,

Granting

the infinite

Thus it is natural

p

Indeed a r e p r e s e n t a t i o n

in the literature.

occurs

r 3.

~.

~ of

every

conjecture

if and only if it occurs

Note that cusp forms

contain

ZA\G~

and later verified

occur in

7 : D(S(~))

discretely

on

of this R a m a n u j a n - P e t e r s s o n

~

implicit

S(~)

(6.1),

v

For this

which asserts

result should also follow from Deligne's

A consequence each

A ~

conjecture

must occur in the regular

in L2(Gp).

in

don't occur in

is that such cusp forms occur in

r I.

This

is

r 3.

129

consistent of

[40];

with Shimura's

cf.

also paragraph

Although that all

"non-trivial"

Then I can prove

Remark 6.4.

1/2

or

that

determine

~-function

generates

2 orB).

"odd" pieces

non-trivial Howe's

a representation

r~(ql, ) _ Tp)

representation

| r~

with

that

the fact that

remarks

(i)

Suppose

quently,

r~

defines

a

of the basic Weil

observation

this way.

as

com-

was

In our setup

this

with p

r* equal to P does or does not equal

should

correspond

"odd" only for

p

to the

equals

~

or

2.

~ (1-e 2~inz) n=l

1/2 whose

(concerning

| r* P suppose r~

belongs

24-th power 3/2

is the discriminant

and this

Note added

is a "trivial"

is consistent

of "non-trivial" constituent

is odd for an odd number

2.34,

(3/19/76).

(private

the nature

to the space

by Observation

Conjecture

rI

with

r*~ = ~3/2"

More

~ ~(~i)

of

original

~3

is a cusp form of weight

particular,

TO .

that

is the cusp form of weight ~3

to

cusp forms was first

| r~

(according

~(z) = eTriz/12

A(z);

at least belong

constituents

must arise

Howe also surmised

Recall

rI

I can show

3/2.

The belief

could

that Dedekind's

or

of

this conjecture

| r* of r I "nonP for all but a non-zero even number of p.

to me by Roger Howe.

rO(ql , Tp)

to prove

Each non-trivial

cusp form of weight

~I

of p. i

the following:

Theorem 6.3.

means

at the bottom

suppose we call a constituent

r*p = rp(ql, T )

representation

quoted

of p. 478 of [38].

constituents

e

if

municated

(C)

I have been unable

More precisely, trivial"

conjecture*

of | r~,

~(-x)

the corresponding

Deligne

communication

of

of

constituents). r I.

p.

= -@(x).

Then if Conse-

theta-series

and Serre have verified

from Deligne).

In

e(~,g-)

Shimura's

130

is identically

zero.

tion that

be odd for at least one

r~

The orthogonal

an automorphic

| r~

group of

of the p-adic

an even number of

p

on the "rational

subgroup"

of

was included

ql

is

of

Z2

O(ql).

embedded

rI

In particular,

must be trivial

diagonal

y)

Theorem 6.3 is con-

of Subsection

in R. Howe's general

theory of the Weil representation.

Thus

to "automorphlc"

2.~.

We note finally that Theorem 6.3 should ultimately a special example

This means

trivial for all but

correspond

sistent with the general philosophy

to exclude

group is just a collection

groups

[~i]

The condi-

Ix: x 2 = I].

(the global representation

constituents

representations

p

"trivial".

already discussed.

form on the orthogonal

of representations

"non-trlvial"

is indeed

e r e = | rp(ql,~ )

the representation (ii)

I.e.,

appear as

(but as yet unpublished)

The proof of the Theorem

sketched below is not particularly

elegant but it's the only one

I know. Proof qf T heoreFl 6.3. of a constituent

of

occuring

are automatically

in

rI

rI

By Theorem a.19 the infinite component

only to prove that if rI

is

or

~i/2

r* = @

* rp

73/2 .

of weight

I.e.,

cusp forms

1/2 or 3/2.

is a non-trivlal

It remains

constituent

of

then the theta-serles

6(g,~)

belongs to

=

Z r*(~)~(~)

L~(SL2(Q)"-. ~ )

for all appropriate

Note first that we can assume or odd according hence

~p,

as

r~

is even or odd.

is odd for at least one

is also odd,

r*(g)@(O)

~(x)

= 0

p o

= 5 ~p(X)

@ 9 with

~p

even r P~

But by hypotheses, Therefore,

and the sum ~n (6.2)

since

r~@

extends only over

~Qx . t For each

gr

write

g=~atk

with

hens,

at=[o

0 t -1]'

131 t cAX

and

k=SO2~'T~~)

there exists some

= K* o We s h a l l prove now that

fo ~ /~QA )

suc~ that

[r*(g){(~])l

(6.B) for all

~=nat~

in

~).

~ I tll/270(t~) Our method of proof is based

on

[36]~

From formulas (2.42) and (2.45) it follows that

Ir*(~)~(~)l = I t I 1 / 2 ( r * ( ~ ) { ) ( t ~ )

(6.~) But the map

(~,~) ~ r(g){

into

.

J(%)

Therefore,

implies there exists for all

~ eg

and

defines a continuous map of since

fo~ ~(QA ) kcK*.

&

and a constant

M

~ (A)

this

such that

Ir*(~)~(~)I ~ fo(~)

f ~ J(A),

there exists a positive t ~A x,

~ MItl -2&

inequality has only to do with the definition of

we omit its straightforward proof.

Together with (6.3)

it implies

(6.5)

Z

~x

r*(g)~(~) ~_ Nit[l/2 -2&

To conclude our proof it suffices to show that oounded on (the finite measure space)

SL2(Q)~x S - ~ )

e(~,g)

is

For this

we appeal to a well known result from reduction theory, namely that t 0 SL2(A ) , SL2(~)NAAc K* with A c = [[0 t_l ]: Itl ~ c]. This result

implies

8-~2(~1 - SL2(Q)N~AcT~* 9

le(E,|

[]

Thus, f o r a l l

Mitl 1/2 ~M'

as was to be shown.

A)

is compact, Lemma 5 of [47]

such that for all

Z Tf(t{)l ~cQx SinCe

~*

~ x ~ ( Q

This, together, with (6.~), proves (6.3).

Next we observe that for any integer

9

,

-2~

g e 8L2(Q)kS-~),

b.3. More open problems Numerous mentioned

loose ends and open problems have already been

in Sections

1 through 6.

to mention two more open problems, The first concerns a constituent is "no"

of

([18],

70

~

I want

both non-trlvial.

a "multiplicity

one" theorem for

occur more than once?

Proposition

two cusp forms on

In this last paragraph

ll.l.1).

For

TO

T 0.

Can

the answer

A related question

is whether

must be equal if they agree at all but

finitely many finite places and share the same central character? For

TO

the answer is "yes".

one theorem"

of Casselman

This is the "strong multiplicity

[4] and Miyake

[27].

In classical

terms

the question is whether a cusp form of given weight is determined by almost all its elgenvalues can associate L0

for the Hecke ring and whether one

to this form a well-deflned

irreducible

subspace

of

9 A second problem is to develop a theory of automorphic

for n-fold covermng groups of

GL2(a ).

foz~s

Such groups were constructed

by Kubota in [20] and by Moore in [29]. It seems clear that most of the results of Section 3 through 5 will extend to these n-fold covering groups without modification

(the singularities

at

series and induced representations, ties at

s = l/n).

s = 1/2, both for Eisensteln are to be replaced by singulari-

In fact the results of Eubota described

Section 1 were originally The fundamental

developed

class determined

order 2.)

Thus some kind of analogue

needed.

Although

initial

by Well's

representation

(The

is a lwaxs of

for this representation

polynomials

steps for

for n-fold covering

is no longer available.

cohomology

to homogeneous

in

by him in this generality.

problem we encounter

groups is that Well's construction

corresponding

essential

of higher degree is

F = ~

Kubota in [21] much work remains to be done.

have been taken by The case of

133

p-adic fields already presents Note added (March 1976). correct two formulas in [i0].

real difficulties. We take this opportunity to The last formula displayed on

p.l~09 should read

M(s,~)

~(2s'~2) ~ M(~,s)~ L(2s+l,~2)

the integral expression above it should read

~(2s+l,(~)2) L(2s, (~$)2)

~N v ~(wng)dn .

References I.

Artin 9

E., and J. Tate,

New York 9 2.

Bass,

H.,

Math. Boerner,

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I.H.E.S.,

H.,

Casselman,

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W.,

6.

Gelbart,

S., Automorphic

Mathematics

Study,

Fourier Number

,

9

Forms

Publ.

North-Holland,

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forms

Math.

Ann.

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Annals

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Cornell

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Memoirs

on the metaplectic

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I.H.E.S.

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Automorphic notes 9

Publ.

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35.

Shalika, J., Representations of the two-by=two unimodular ~roup 9vet local fields , Thesis, Jol~ns Hopkins University, 1966.

36.

, and S. Tanaka, On an explicit construction of a certain class of autemorphic form_~s, Amer. J. o:f Math. 91 (1969), pp. 1043-1076.

37.

Shimizu, H., Theta series and automorpbic forms on

GL(2),

Journ. Math. Soc. Japan, 24 (1972), pp. 638-683. 38.

Shimura, G., On modular forms of half-integral weight, Ann. of Math. 97 (1973), pp. 440-481.

39.

, On the holomorlhy of certain Dirichlet series, Proc. London Math. Soc (3) 31(1975).

~0.

Shimura, G., 0n the t rac@ formula for Hecke operators, Acta Math., 132 (197h)o

41.

Shintani, T., On construction of holomorphic cusp forms of half-integral weight, Nagoya J. of Math., 58 (1975).

42.

Siegel, C.L., Die FunktionaI gleichungen eineger Dirichletscher Reihen, Math. Zeit. 63 (1956), pp. 363-373.

43.

Strichartz, R., F0urier translf0~ s and non-compact rotation groups,

h4.

Indiana U. Math.

J.(6) 24(]974).

,, Harmonic analysis on hy~erboloids,

J. Functional

Anal., 12 (1973), pp. 341-383. 45.

Tanaka, S., On irreducible unitary representations of some special linear grou~s of the second order, Osaka J. Math., vol. 3, PP. 217-227.

137

46.

Tare, J., Fou
in Algebraic Number Theory, Tompson Book Co.,

Washington, %7.

D. C. (1967).

Well, A., Sur certaines ~roup.es d'o~erateurs

un~taires,

Acta

Math. iii (196~), pp. 1%3-911. %8.

, Sur la formule de Sieg_~l dans ia th~ori@ des ~rou~es classiques,

Acta Math. 113 (1967).

The following papers are also relevant to the subject matter of these Notes: %9.

Asai, T., The reciprocity of Dedek~nd s~ims and the factor set for the universal covering Group of of Math.,

50.

SL(2,R), Nagoya J.

37 (1970).

Lipsman, R., On the unitary representation of a semi-simple Lie ~roup given by the invariant inte@ral on its Lie a l~ebra, Technical Report TR 74-64, October,

51.

Rallis,

1974.

S. and Schiffman,

groupe ortho~onal,

G., Distributions

preprint

Ranga Rao, R., On the Well representation a quadratic form,

53.

Ehrenpreis,

Providence,

(1974).

of SL 2 associated to

L., The use of partial differential

Symposia in Pure Mathematics,

Gelbart,

preprint

in preparation.

the study of ~roup representations,

54.

invariantes par le

(1974); also, Well representa-

tion I: Intertwlnin~ distributions, 52.

University of Maryland,

equations for

in Proceedings of

A.M.S. Volume 26(197%),

R.I.

S., and Jacquet, H., On the symmetric square of an

automorphic form,

in preparation.

Subject Index Abstract symplectic group, -representation,

32

29

aut omorphic forms: - on GL(2),

6

- o f half-integral weight,

5, 6, 51, 58

Class i representations : - definition,

!03

- characterization of, Clifford theory,

103

78, 98

Complementary series representation, continuous series representation,

77, 114

77, 114-115

cusp forms: -

classical, of weight k/2, 52, 65, 67

- g e n e r a l i z e d ,

- construction of, D-duality correspondence, dyadic fields,

i, 51

125 6, 39

101

Eisenstein series: - for imaginary quadratic fields, - for GL(2),

4

63

- for the metaplectic groupd, -

poles of,

-

representations generated by,

65

64, 67 63, 67, 123

even piece of the basic Weil representation, (archimedean case), factor set,

115

11

factorizability, genuine functions,

59 6

genuine representations, group extensions,

llff.

29

94-95

(non-archimedean),

122 (global)

139

~(G,T),

11

Hecke operators: classical,

-

2, 106

- for the metaplectic group, Heisenberg group, Hilbert symbol,

105

31 13

local map: - for archimedean places,

81

- for non-archimedean places,

ili-115

metaplectic group: - local,

14ff.

global,

-

5, 22

- Kubota's construction of, -

Well's construction of,

multiplier,

13 36

11

multiplier representation,

29

odd piece of the basic Well representation, 94-95 (archimedean), 118 (non-archimedean), 128 (global) principal series representation, projective representations, pseudo-symplectic group,

69

28

34

Ramanu~an-Petersson Conjecture,

128

quadratic power residue symbol,

24

representations of ~2(F), - characters of, -

class i,

96

11!

102

- principal series, representations of

97

G-L2(R), 79

representations of S-L2(R): - complementary series,

77

140

representations of

SL2(R):

discrete series,

-

- irreducible,

77

77

principal series,

-

"trash",

-

73

78

self-conjugate representations, special representation,

78

114-115

spectrum of the metaplectic group,

5, 62, 121:

- continuous, 121 - discrete,

121

- discrete non-cuspidal,

122

strong multiplicity one theorem, supercuspidal representation, theta-functions,

78

11

Weil's representation,

28-38

-cor~nuting algebra of, -

96, 118

46, 128

trash representation, two-cocyele,

extension to

GL(2),

39 41

- explicit description of, -

-

126, 132

in 3-variables, in l-variable,

86 (real case), 126 (global) 93 (real), 118 (non-archimedean),

i22 (global) - philosophy of,

36

39